TheoremsDefinitionsProof MethodsProblem Solving Techniques

This is a catalog of theorems from Intermediate Analysis, alongside proofs for each.

Theorem 1. 5.1.9 Sequential Criterion for Limits
Let f:D→Rand let c be an accumulation point of D.Then lim⁡x→cf(x)=Liff for every sequence (sn)∈Dthat converges to c with sn≠c  ∀n∈N,the sequence (f(sn)) converges to L.\text{Let } f: D \to \R \\ \text{and let c be an accumulation point of D.} \\ \text{Then } \lim_{x \to c} f(x) = L \\ \text{iff for every sequence } (s_n) \in D \\ \text{that converges to } c \text{ with } s_n \neq c \ \ \forall n \in \N, \\ \text{the sequence } (f(s_n)) \text{ converges to } L.

Theorem 2. 5.1.14 Operations on Limits are Well-Defined
Let f:D→R, and c be an accumulation point of D.If lim⁡x→cf(x)=L, lim⁡x→cg(x)=M,and k∈R, then the following hold: a. lim⁡x→c(f+g)(x)=L+M. b. lim⁡x→c(fg)(x)=LM. c. lim⁡x→c(kf)(x)=kL. d. If g(x)≠0  ∀x∈D and M≠0,    then lim⁡x→c(fg)(x)=LM\text{Let } f: D \to \R, \text{ and } c \text{ be an accumulation point of } D. \\ \text{If } \lim_{x \to c} f(x) = L, \ \lim_{x \to c} g(x) = M, \\ \text{and } k \in \R, \text{ then the following hold:} \\ \ \\ \text{a. } \lim_{x \to c} (f+g)(x) = L + M. \\ \ \\ \text{b. } \lim_{x \to c} (fg)(x) = LM. \\ \ \\ \text{c. } \lim_{x \to c} (kf)(x) = kL. \\ \ \\ \text{d. If } g(x) \neq 0 \ \ \forall x \in D \text{ and } M \neq 0, \\ \ \ \ \ \text{then } \lim_{x \to c} \left(\frac{f}{g}\right)(x) = \frac{L}{M}

Theorem 3. 5.1.19 Sides of a Limit Must Match
Let f be a function defined on a deleted neighborhoodof a point c. Then lim⁡x→cf(x)=Liff lim⁡x→c+f(x)=lim⁡x→c−f(x)\text{Let } f \text{ be a function defined on a deleted neighborhood} \\ \text{of a point } c. \text{ Then } \lim_{x \to c} f(x) = L \\ \text{iff } \lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)

Theorem 4. 5.2.3(a,b,d) Sequential Criterion for Continuity
Letf:D→R and let c∈D.Then the following conditions are equivalent: a. f is continuous at c. b. If (xn) is any sequence in D such that (xn)→c,    then lim⁡n→∞f(xn)=f(c). Furthermore, if c is an accumulation point of D,then the above are equivalent to: d. f has a limit at c and lim⁡x→cf(x)=f(c).\text{Let} f: D \to \R \text{ and let } c \in D. \\ \text{Then the following conditions are equivalent:} \\ \ \\ \text{a. } f \text{ is continuous at c.} \\ \ \\ \text{b. If } (x_n) \text{ is any sequence in } D \text{ such that } (x_n) \to c, \\ \ \ \ \ \text{then } \lim_{n \to \infty} f(x_n) = f(c). \\ \ \\ \text{Furthermore, if } c \text{ is an accumulation point of } D, \\ \text{then the above are equivalent to:} \\ \ \\ \text{d. } f \text{ has a limit at } c \text{ and } \lim_{x \to c} f(x) = f(c).

Theorem 5. 5.2.7 Sequential Method for Discontinuity
Let f:D→R and let c∈D.Then f is discontinuous at c iffthere exists a sequence (xn) in Dsuch that (xn)→cbut the sequence (f(xn))↛f(c).\text{Let } f: D \to \R \text{ and let } c \in D. \\ \text{Then } f \text{ is discontinuous at } c \text{ iff} \\ \text{there exists a sequence } (x_n) \text{ in } D \\ \text{such that } (x_n) \to c \\ \text{but the sequence } (f(x_n)) \not\to f(c).

Theorem 6. 5.2.11 Operations on Functions Preserve Continuity
Let f:D→R and g:D→R and let c∈D.Suppose that f and g are continuous at c.Then:  a. f+g is continuous at c, b. fg is continuous at c, c. fg continuous at c if g(c)≠0. \text{Let } f: D \to \R \text{ and } g: D \to \R \text{ and let } c \in D. \\ \text{Suppose that } f \text{ and } g \text{ are continuous at } c. \\ \text{Then: } \\ \ \\ \text{a. } f + g \text{ is continuous at } c, \\ \ \\ \text{b. } fg \text{ is continuous at } c, \\ \ \\ \text{c. } \frac{f}{g} \text{ continuous at } c \text{ if } g(c) \neq 0.

Theorem 7. 5.2.13 Composition Preserves Continuity
Let f:D→R and g:E→Rbe functions such thatf(D)⊆E.If f is continuous at a point c∈Dand g is continuous at f(c),then the composition g∘f:D→Ris continuous at c.\text{Let } f: D \to \R \text{ and } g: E \to \R \\ \text{be functions such that} f(D) \subseteq E. \\ \text{If } f \text{ is continuous at a point } c \in D \\ \text{and } g \text{ is continuous at } f(c), \\ \text{then the composition } g \circ f: D \to \R \\ \text{is continuous at } c.

Theorem 8. 5.3.3 Image of Compact Subsets are Compact Subsets
Let D be a compact subset of Rand suppose that f:D→R is continuous.Then f(D) is compact.\text{Let } D \text{ be a compact subset of } \R \\ \text{and suppose that } f: D \to \R \text{ is continuous.} \\ \text{Then } f(D) \text{ is compact.}

Theorem 9. 5.3.4 Extreme Value Theorem
Let D be a compact subset of Rand suppose that f:D→R is continuous.Then f assumes minimum and maximum values on D.That is,  ∃x1,x2∈Dsuch that f(x1)≤f(x)≤f(x2)  ∀x∈D.\text{Let } D \text{ be a compact subset of } \R \\ \text{and suppose that } f: D \to \R \text{ is continuous.} \\ \text{Then } f \text{ assumes minimum and maximum values on } D. \\ \text{That is, } \ \exists x_1, x_2 \in D \\ \text{such that } f(x_1) \leq f(x) \leq f(x_2) \ \ \forall x \in D.

Theorem 10. 5.3.7 Intermediate Value Theorem
Suppose that f:[a,b]→R is continuous.Then f has the intermediate value property on [a,b].That is, if k is any value between f(a) and f(b)(f(a)<k<f(b) or f(b)<k<f(a)),then  ∃c∈(a,b) such that f(c)=k.\text{Suppose that } f: [a,b] \to \R \text{ is continuous.} \\ \text{Then } f \text{ has the intermediate value property on [a,b].} \\ \text{That is, if } k \text{ is any value between } f(a) \text{ and } f(b) \\ (f(a) < k < f(b) \text{ or } f(b) < k < f(a)), \\ \text{then } \ \exists c \in (a,b) \text{ such that } f(c) = k.

Theorem 11. 5.4.6 Continuity on a Compact Set Implies UC
Suppose f:D→R is continuous on a compact set D.Then f is uniformly continuous on D.\text{Suppose } f: D \to \R \text{ is continuous on a compact set } D. \\ \text{Then } f \text{ is uniformly continuous on } D.

Theorem 12. ★ 5.4.8 UC Image of a Cauchy Sequence is Cauchy
Let f:D→R be uniformly continuous on Dand suppose that (xn) is a Cauchy sequence in D.Then (f(xn)) is a Cauchy sequence.\text{Let } f: D \to \R \text{ be uniformly continuous on } D \\ \text{and suppose that } (x_n) \text{ is a Cauchy sequence in } D. \\ \text{Then } (f(x_n)) \text{ is a Cauchy sequence}.

Theorem 13. 6.1.3 Sequential Condition for Derivatives
Let I be an interval with c∈I, and f:I→R. Then f is differentiable at c iff,for every sequence (xn)∈I such that (xn)→c and xn≠c  ∀n, the sequence (f(xn)−f(c)xn−c) converges. Furthermore, if f is differentiable at c,then the sequence of quotients will converge to f′(c).\text{Let } I \text{ be an interval with } c \in I, \text{ and } f: I \to \R. \\ \ \\ \text{Then } f \text{ is differentiable at } c \text{ iff}, \\ \text{for every sequence } (x_n) \in I \text{ such that } (x_n) \to c \text{ and } x_n \neq c \ \ \forall n, \\ \ \\ \text{the sequence } \left(\frac{f(x_n)-f(c)}{x_n-c}\right) \text{ converges.} \\ \ \\ \text{Furthermore, if } f \text{ is differentiable at } c, \\ \text{then the sequence of quotients will converge to } f'(c).

Theorem 14. ★ 6.1.6 Differentiability Implies Continuity
If f:I→R is differentiable at c∈I,then f is continuous at c.\text{If } f: I \to \R \text{ is differentiable at } c \in I, \\ \text{then } f \text{ is continuous at } c.

Theorem 15. 6.1.7 Derivative Rules
Suppose that f:I→R and g:I→Rare differentiable at c∈I.Then a. If k∈R, then kf is differentiable at c and     (kf)′(c)=k⋅f′(c). b. The function f+g is diffterentiable at c and     (f+g)′(c)=f′(c)+g′(c). c. (Product Rule) The function fg is differentiable at c and     (fg)′(c)=f(c)g′(c)+g(c)f′(c). d. (Quotient Rule) If g(c)≠0, then fg is differentiable at c and     (fg)′(c)=g(c)f′(c)−f(c)g′(c)[g(c)]2\text{Suppose that } f: I \to \R \text{ and } g: I \to \R \\ \text{are differentiable at } c \in I. Then \\ \ \\ \text{a. If } k \in \R, \text{ then } kf \text{ is differentiable at } c \text{ and} \\ \ \\ \ \ \ \ (kf)'(c) = k \cdot f'(c). \\ \ \\ \text{b. The function } f + g \text{ is diffterentiable at } c \text{ and} \\ \ \\ \ \ \ \ (f + g)'(c) = f'(c) + g'(c). \\ \ \\ \text{c. (Product Rule) The function } fg \text{ is differentiable at } c \text{ and} \\ \ \\ \ \ \ \ (fg)'(c) = f(c)g'(c) + g(c)f'(c). \\ \ \\ \text{d. (Quotient Rule) If } g(c) \neq 0, \text{ then } \frac{f}{g} \text{ is differentiable at } c \text{ and} \\ \ \\ \ \ \ \ \left(\frac{f}{g}\right)'(c) = \frac{g(c)f'(c)-f(c)g'(c)}{[g(c)]^2}

Theorem 16. 6.1.10 Chain Rule
Let I,J be intervals in R, let f:I→R and g:J→R,where f(I)⊆J, and let c∈I. If f is differentiable at c and g is differentiable at f(c),then g∘f is differentiable at c and (g∘f)′(c)=g′(f(c))⋅f′(c).\text{Let } I, J \text{ be intervals in } \R, \text{ let } f: I \to \R \text{ and } g: J \to \R, \\ \text{where } f(I) \subseteq J, \text{ and let } c \in I. \\ \ \\ \text{If } f \text{ is differentiable at } c \text{ and } g \text{ is differentiable at } f(c), \\ \text{then } g \circ f \text{ is differentiable at } c \text{ and} \\ \ \\ (g \circ f)'(c) = g'(f(c)) \cdot f'(c).

Theorem 17. ★ 6.2.2 Peaks and Troughs
If f is differentiable on an open interval (a,b)and if f achieves its absolute max or min at c∈(a,b),then f′(c)=0.\text{If } f \text{ is differentiable on an open interval } (a,b) \\ \text{and if } f \text{ achieves its absolute max or min} \text{ at } c \in (a,b), \\ \text{then } f'(c) = 0.

Theorem 18. 6.2.4 Rolle's Theorem
Let f be a continuous function on [a,b]that is differentiable on (a,b)and such that f(a)=f(b).Then ∃c∈(a,b) such that f′(c)=0.\text{Let } f \text{ be a continuous function on } [a,b] \\ \text{that is differentiable on } (a, b) \\ \text{and such that } f(a) = f(b). \\ \text{Then } \exists c \in (a,b) \text{ such that } f'(c) = 0.

Theorem 19. 6.2.5 Mean Value Theorem
Let f be a continuous function on [a,b]that is differentiable on (a,b). Then ∃c∈(a,b) such that  f′(c)=f(b)−f(a)b−a.\text{Let } f \text{ be a continuous function on } [a,b] \\ \text{that is differentiable on } (a,b). \\ \ \\ \text{Then } \exists c \in (a,b) \text{ such that } \\ \ \\ f'(c) = \frac{f(b)-f(a)}{b-a}.

Theorem 20. 6.2.8 Constant Functions
Let f be continuous on [a,b] and differentiable on (a,b).If f′(x)=0  ∀x∈(a,b),then f is constant on [a,b].\text{Let } f \text{ be continuous on } [a,b] \text{ and differentiable on } (a,b). \\ \text{If } f'(x) = 0 \ \ \forall x \in (a,b), \\ \text{then } f \text{ is constant on } [a,b].

Theorem 21. 6.2.9 Same Derivative Implies Antiderivatives Offset by a Constant
Let f and g be continuous on [a,b] and differentiable on (a,b).Suppose that f′(x)=g′(x)  ∀x∈(a,b),then ∃ a constant C such that f=g+C on [a,b].\text{Let } f \text{ and } g \text{ be continuous on } [a,b] \text{ and differentiable on } (a,b). \\ \text{Suppose that } f'(x) = g'(x) \ \ \forall x \in (a,b), \\ \text{then } \exists \text{ a constant } C \text{ such that } f = g + C \text{ on } [a,b].

Theorem 22. ★ 6.2.11 Strict Parity Implies Monotonicity
Let f be differentiable on an interval I. Then a. If f′(x)>0  ∀x∈I,    then f is strictly increasing on I. b. If f′(x)<0  ∀x∈I,    then f is strictly decreasing on I.\text{Let } f \text{ be differentiable on an interval } I. \text{ Then} \\ \ \\ \text{a. If } f'(x) > 0 \ \ \forall x \in I, \\ \ \ \ \ \text{then } f \text{ is strictly increasing on } I. \\ \ \\ \text{b. If } f'(x) < 0 \ \ \forall x \in I, \\ \ \ \ \ \text{then } f \text{ is strictly decreasing on } I.

Theorem 23. 6.2.13 Inverse Function Theorem
Suppose that f is differentiable on an interval Iand f′(x)≠0  ∀x∈I. Then f is injective, f−1 is differentiable on f(I), and  (f−1)′(y)=1f′(x), where y=f(x).\text{Suppose that } f \text{ is differentiable on an interval } I \\ \text{and } f'(x) \neq 0 \ \ \forall x \in I. \\ \ \\ \text{Then } f \text{ is injective, } \\ f^{-1} \text{ is differentiable on } f(I), \text{ and } \\ \ \\ (f^{-1})'(y) = \frac{1}{f'(x)}, \\ \ \\ \text{where } y = f(x).

Theorem 24. 7.1.5 Refinements of Partitions have More Accurate Darboux Sums
Let f be a bounded function on [a, b].If P and Q are partiitions of [a,b]and Q is a refinement of P, thenL(f,P)≤L(f,Q)≤U(f,Q)≤U(f,P).\text{Let } f \text{ be a bounded function on [a, b].} \\ \text{If } P \text{ and } Q \text{ are partiitions of } [a,b] \\ \text{and } Q \text{ is a refinement of } P, \text{ then} \\ L(f, P) \leq L(f, Q) \leq U(f, Q) \leq U(f, P).

Theorem 25. 7.1.7 Lower Sum Below Upper Sum
Let f be a bounded function on [a,b]. Then L(f)≤U(f).\text{Let } f \text{ be a bounded function on } [a,b]. \text{ Then } L(f) \leq U(f).

Theorem 26. 7.1.10 Criterion for Integrability
Let f be a bounded function on [a,b].Then f is integrable iff for each ε>0∃ a partition P of [a,b] such thatU(f,P)−L(f,P)<ε.\text{Let } f \text{ be a bounded function on } [a,b]. \\ \text{Then } f \text{ is integrable iff for each } \varepsilon > 0 \\ \exists \text{ a partition } P \text{ of } [a,b] \text{ such that} \\ U(f, P) - L(f, P) < \varepsilon.

Theorem 27. ★ 7.2.1 Monotone Implies Integrable
Let f be a monotone function on [a,b]. Then f is integrable.\text{Let f be a monotone function on } [a,b]. \text{ Then } f \text{ is integrable.}

Theorem 28. ★ 7.2.2 Continuous Implies Integrable
Let f be a continuous function on [a,b]. Then f is integrable on [a,b].\text{Let } f \text{ be a continuous function on } [a,b]. \text{ Then } f \text{ is integrable on } [a, b].

Theorem 29. 7.2.4 Linearity of The Integral
Let f and g be integrable functions on [a,b] and let k∈R. Then a. kf is integrable and ∫abkf=k∫abf, and b. f+g is integrable and ∫ab(f+g)=∫abf+∫abg.\text{Let } f \text{ and } g \text{ be integrable functions on } [a,b] \text{ and let } k \in \R. \text{ Then} \\ \ \\ \text{a. } kf \text{ is integrable and } \int_a^b kf = k \int_a^b f, \text{ and} \\ \ \\ \text{b. } f+g \text{ is integrable and } \int_a^b (f+g) = \int_a^b f + \int_a^b g.

Theorem 30. 7.2.6 Split Bounds of Integral
Suppose that f is integrable on both [a,c] and [c,b]. Then f is integrable on [a,b]. Furthermore, ∫abf=∫acf+∫cbf.\text{Suppose that } f \text{ is integrable on both } [a,c] \text{ and } [c, b]. \\ \ \\ \text{Then } f \text{ is integrable on } [a,b]. \\ \ \\ \text{Furthermore, } \int_a^b f = \int_a^c f + \int_c^b f.

Theorem 31. 7.2.8 A Triangle Inequality for The Integral
Let f be integrable on [a,b]. Then ∣f∣ is integrable on [a,b] and∣∫abf∣≤∫ab∣f∣.\text{Let } f \text{ be integrable on } [a,b]. \\ \text{ Then } |f| \text{ is integrable on } [a,b] \text{ and} \\ \left| \int_a^b f \right| \leq \int_a^b |f|.

Theorem 32. ★ 7.3.1 Fundamental Theorem of Calculus 1
Let f be integrable on [a,b].  ∀x∈[a,b], let F(x)=∫axf(t)dt. Then F is uniformly continuous on [a,b]. Furthermore, if f is continuous at c∈[a,b],then F is differentiable at c and F′(c)=f(c).\text{Let } f \text{ be integrable on } [a,b]. \ \ \forall x \in [a,b], \text{ let } F(x) = \int_{a}^{x} f(t) dt. \\ \ \\ \text{Then } F \text{ is uniformly continuous on } [a,b]. \\ \ \\ \text{Furthermore, if } f \text{ is continuous at } c \in [a,b], \\ \text{then } F \text{ is differentiable at } c \text{ and } F'(c) = f(c).

Theorem 33. 7.3.3 Chain Rule for FTC 1
Let f be continuous on [a,b]and let g be differentiable on [c,d],where g([c,d])⊆[a,b]. Define F(x)=∫ag(x)f, for all x∈[a,b] Then F is differentiable on [c,d] and F′(x)=[f∘g(x)]⋅g′(x)\text{Let } f \text{ be continuous on } [a,b] \\ \text{and let } g \text{ be differentiable on } [c,d], \text{where } g([c,d]) \subseteq [a,b]. \\ \ \\ \text{Define } F(x) = \int_a^{g(x)} f, \text{ for all } x \in [a,b] \\ \ \\ \text{Then } F \text{ is differentiable on } [c,d] \text{ and } F'(x) = [f \circ g(x)]\cdot g'(x)

Theorem 34. ★ 7.3.5 Fundamental Theorem of Calculus 2
If f is differentiable on [a,b] and f′ is integrable on [a,b], then ∫abf′=f(b)−f(a)\text{If } f \text{ is differentiable on } [a,b] \text{ and } f' \text{ is integrable on [a,b],} \\ \ \\ \text{then } \int_a^b f' = f(b) - f(a)

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