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Catalog of Definitions from Intermediate Analysis.

Definition 1. 5.1.1 Limit
Let f:D→R and let c be an accumulation point of D.We say that a real number L is a limit of f at c, if for each ε>0 there exists a δ>0such that ∣f(x)−L∣<ε whenever x∈Dand 0<∣x−c∣<δ.\text{Let } f: D \to \R \text{ and let } c \text{ be an accumulation point of D.} \\ \text{We say that a real number } L \text{ is a limit of } f \text{ at } c, \text{ if} \\ \ \\ \quad \text{for each } \varepsilon \gt 0 \text{ there exists a } \delta \gt 0 \\ \quad \text{such that } |f(x) - L| < \varepsilon \text{ whenever } x \in D \\ \quad \text{and } 0 < |x-c| < \delta.
Definition 2. 5.1.13 Operations on Function
Let f:D→R and g:D→R. a. The sum f+g:D→R    is given by (f+g)(x)=f(x)+g(x)  ∀x∈D. b. The product fg:D→R    is given by (fg)(x)=f(x)∗g(x)  ∀x∈D. c. For k∈R, the multiple kf:D→R    is given by (kf)(x)=k∗f(x)  ∀x∈D. d. If g(x)≠0 ∀x∈D, the quotient f/g:D→R    is given by (fg)(x)=f(x)g(x)  ∀x∈D.\text{Let } f: D \to \R \text{ and } g: D \to \R. \\ \ \\ \quad \text{a. The sum } f + g: D \to \R \\ \quad \ \ \ \ \text{is given by } (f+g)(x) = f(x)+g(x) \ \ \forall x \in D. \\ \ \\ \quad \text{b. The product } fg: D \to \R \\ \quad \ \ \ \ \text{is given by } (fg)(x) = f(x)*g(x) \ \ \forall x \in D. \\ \ \\ \quad \text{c. For } k \in \R, \text{ the multiple } kf: D \to \R \\ \quad \ \ \ \ \text{is given by } (kf)(x) = k*f(x) \ \ \forall x \in D. \\ \ \\ \quad \text{d. If } g(x) \neq 0 \ \forall x \in D, \text{ the quotient } f/g: D \to \R \\ \quad \ \ \ \ \text{is given by } \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \ \ \forall x \in D.
Definition 3. 5.1.18 Right and Left Limit
Let f:(a,b)→R. a. We say that a real number L    is the right-hand limit of f at a and write    lim⁡x→a+f(x)=L,    if  ∀ε>0  ∃δ>0 such that ∣f(x)−L∣<ε    whenever x∈(a,b) and a<x<a+δ. b. We say that a real number L    is the left-hand limit of f at a and write    lim⁡x→a−f(x)=L,    if  ∀ε>0  ∃δ>0 such that ∣f(x)−L∣<ε    whenever x∈(a,b) and a−δ<x<a.\text{Let } f: (a, b) \to \R. \\ \ \\ \quad \text{a. We say that a real number } L \\ \quad \ \ \ \ \text{is the right-hand limit of } f \text{ at } a \text{ and write} \\ \quad \ \ \ \ \lim_{x \to a^+}f(x) = L, \\ \quad \ \ \ \ \text{if } \ \forall \varepsilon > 0 \ \ \exists \delta > 0 \text{ such that } |f(x) - L| < \varepsilon \\ \quad \ \ \ \ \text{whenever } x \in (a, b) \text{ and } a < x < a + \delta. \\ \ \\ \quad \text{b. We say that a real number } L \\ \quad \ \ \ \ \text{is the left-hand limit of } f \text{ at } a \text{ and write} \\ \quad \ \ \ \ \lim_{x \to a^-}f(x) = L, \\ \quad \ \ \ \ \text{if } \ \forall \varepsilon > 0 \ \ \exists \delta > 0 \text{ such that } |f(x) - L| < \varepsilon \\ \quad \ \ \ \ \text{whenever } x \in (a, b) \text{ and } a - \delta < x < a.
Definition 4. 5.2.1 Continuous
Let f:D→R and let c∈D. a. We say that f is continuous at c if      ∀ε>0  ∃δ>0 such that ∣f(x)−f(c)∣<ε     whenever x∈D and ∣x−c∣<δ. b. If f is continuous at each point of a subset S⊆D,    then f is said to be continuous on S. c. If f is continuous on its domain D,    Then f is said to be a continuous function.\text{Let } f: D \to \R \text{ and let } c \in D. \\ \ \\ \quad \text{a. We say that } f \text{ is continuous at } c \text{ if} \\ \ \\ \quad \ \ \ \ \ \forall \varepsilon > 0 \ \ \exists \delta > 0 \text{ such that } |f(x) - f(c)| < \varepsilon \\ \quad \ \ \ \ \ \text{whenever } x \in D \text{ and } |x-c| < \delta. \\ \ \\ \quad \text{b. If } f \text{ is continuous at each point of a subset } S \subseteq D, \\ \quad \ \ \ \ \text{then } f \text{ is said to be continuous on S.} \\ \ \\ \quad \text{c. If } f \text{ is continuous on its domain } D, \\ \quad \ \ \ \ \text{Then } f \text{ is said to be a continuous function.}
Definition 5. 5.3.1 Boundedness of a Function
A function f:D→R is said to be boundedif its range f(D) is a bounded subset of R.That is, f is bounded if  ∃M∈Rsuch that ∣f(x)∣≤M  ∀x∈D.\text{A function } f: D \to \R \text{ is said to be bounded} \\ \text{if its range } f(D) \text{ is a bounded subset of } \R. \\ \text{That is, } f \text{ is bounded if } \ \exists M \in \R \\ \text{such that } |f(x)| \leq M \ \ \forall x \in D.
Definition 6. 5.4.1 Uniformly Continuous
Let f:D→R.We say that f is uniformly continuous on D if ∀ε>0  ∃δ>0 such that∣f(x)−f(y)∣<εwhenever ∣x−y∣<δ and x,y∈D.\text{Let } f: D \to \R. \\ \text{We say that } f \text{ is uniformly continuous on } D \text{ if} \\ \ \\ \quad \forall \varepsilon > 0 \ \ \exists \delta > 0 \text{ such that} |f(x) - f(y)| < \varepsilon \\ \quad \text{whenever } |x-y| < \delta \text{ and } x, y \in D.
Definition 7. 6.1.1 Differentiable
Let f be a real-valued function defined on an interval Icontaining the point c. We say that f is differentiable at cif the limit  lim⁡x→cf(x)−f(c)x−c exists and is finite.  We denote the derivative of f at c by f′(c)=lim⁡x→cf(x)−f(c)x−c\text{Let } f \text{ be a real-valued function defined on an interval } I \\ \text{containing the point c. We say that } f \text{ is differentiable at } c \\ \text{if the limit } \\ \ \\ \lim_{x \to c} \frac{f(x)-f(c)}{x-c} \text{ exists and is finite. } \\ \ \\ \text{We denote the derivative of } f \text{ at } c \text{ by} \\ \ \\ f'(c) = \lim_{x \to c} \frac{f(x)-f(c)}{x-c}
Definition 8. 6.2.1 Absolute and Local Maxima
Let f:D→R be a function. a. f has an absolute maximum (minimum) at    c∈D if f(c)≥f(x) (f(c)≤f(x))  ∀x∈D. b. f has a local maximum (minimum)at c∈D    if there exists and open interval I containing c    such that f(c)≥f(x) (f(c)≤f(x))  ∀x∈I∩D.\text{Let } f: D \to \R \text{ be a function.} \\ \ \\ \text{a. } f \text{ has an absolute maximum (minimum) at} \\ \ \ \ \ c \in D \text{ if } f(c) \geq f(x) \ (f(c) \leq f(x)) \ \ \forall x \in D. \\ \ \\ \text{b. } f \text{ has a local maximum (minimum)} \text{at } c \in D \\ \ \ \ \ \text{if there exists and open interval } I \text{ containing c} \\ \ \ \ \ \text{such that } f(c) \geq f(x) \ (f(c) \leq f(x)) \ \ \forall x \in I \cap D.
Definition 9. 6.2.10 Monotonicity of Functions
Let f be a function and I be an interval. a. f is increasing on I if f(x1)≤f(x2)    for all x1,x2∈I with x1<x2. b. f is decreasing on I if f(x1)≥f(x2)    for all x1,x2∈I with x1<x2. c. f is strictly increasing on I if f(x1)<f(x2)    for all x1,x2∈I with x1<x2. d. f is strictly decreasing on I if f(x1)>f(x2)    for all x1,x2∈I with x1<x2.\text{Let } f \text{ be a function and } I \text{ be an interval.} \\ \ \\ \text{a. } f \text{ is increasing on } I \text{ if } f(x_1) \leq f(x_2) \\ \ \ \ \ \text{for all } x_1, x_2 \in I \text{ with } x_1 < x_2. \\ \ \\ \text{b. } f \text{ is decreasing on } I \text{ if } f(x_1) \geq f(x_2) \\ \ \ \ \ \text{for all } x_1, x_2 \in I \text{ with } x_1 < x_2. \\ \ \\ \text{c. } f \text{ is strictly increasing on } I \text{ if } f(x_1) < f(x_2) \\ \ \ \ \ \text{for all } x_1, x_2 \in I \text{ with } x_1 < x_2. \\ \ \\ \text{d. } f \text{ is strictly decreasing on } I \text{ if } f(x_1) > f(x_2) \\ \ \ \ \ \text{for all } x_1, x_2 \in I \text{ with } x_1 < x_2.
Definition 10. 7.1.1 Partition and Refinement
Let [a,b] be an interval in R.A partition P of [a,b] is a finite set of points{x0,x1,...,xn} in [a,b] such that a=x0<x1<...<xn=b. If P and Q are two partitions of [a,b] with P⊆Q,then Q is called a refinement of P.\text{Let } [a,b] \text{ be an interval in } \R. \\ \text{A partition } P \text{ of } [a,b] \text{ is a finite set of points} \\ \{ x_0, x_1, ..., x_n \} \text{ in } [a,b] \text{ such that } a = x_0 < x_1 < ... < x_n = b. \\ \ \\ \text{If } P \text{ and } Q \text{ are two partitions of } [a,b] \text{ with } P \subseteq Q, \\ \text{then } Q \text{ is called a refinement of } P.
Definition 11. 7.1.2 Upper and Lower Sums
Suppose that f is a bounded function defined on [a,b]and that P={x0,x1,...,xn} is a partition of [a,b]. a. For each i∈{1,...,n}, we define    Mi(f)=sup⁡{f(x):x∈[xi−1,xi]},    mi(f)=inf⁡{f(x):x∈[xi−1,xi]},    and Δxi=xi−xi−1 b. The upper sum of f with respect to P is    U(f,P)=∑i=1nMiΔxi. c. The lower sum of f with respect to P is    L(f,P)=∑i=1nmiΔxi.\text{Suppose that } f \text{ is a bounded function defined on } [a,b] \\ \text{and that } P = \{ x_0, x_1, ..., x_n \} \text{ is a partition of } [a,b]. \\ \ \\ \text{a. For each } i \in \{1, ..., n \}, \text{ we define} \\ \ \ \ \ M_i(f) = \sup \{f(x): x \in [x_{i-1}, x_i] \}, \\ \ \ \ \ m_i(f) = \inf \{f(x): x \in [x_{i-1}, x_i] \}, \\ \ \ \ \ \text{and } \Delta x_i = x_i - x_{i-1} \\ \ \\ \text{b. The upper sum of } f \text{ with respect to } P \text{ is} \\ \ \ \ \ U(f,P) = \sum_{i=1}^{n}M_i \Delta x_i. \\ \ \\ \text{c. The lower sum of } f \text{ with respect to } P \text{ is} \\ \ \ \ \ L(f,P) = \sum_{i=1}^{n}m_i \Delta x_i.
Definition 12. 7.1.3 Upper and Lower Integrals
Let f be a bounded function defined on [a,b]. Then a. The upper integral of f on [a,b] is    U(f)=inf⁡{U(f,P):P is a partition of [a,b]}. b. The lower integral of f on [a,b] is    L(f)=sup⁡{L(f,P):P is a partition of [a,b]}.\text{Let } f \text{ be a bounded function defined on } [a,b]. \text{ Then} \\ \ \\ \text{a. The upper integral of } f \text{ on } [a,b] \text{ is} \\ \ \ \ \ U(f) = \inf \{U(f, P): P \text{ is a partition of } [a,b] \}. \\ \ \\ \text{b. The lower integral of } f \text{ on } [a,b] \text{ is} \\ \ \ \ \ L(f) = \sup \{L(f, P): P \text{ is a partition of } [a,b] \}.
Definition 13. 7.1.4 Riemann Integral
Let f be a bounded function defined on [a,b]. Then if L(f)=U(f),we say that f is Riemann integrable on [a,b] and that ∫abf=∫abf(x)dx=L(f)=U(f) is the Riemann integral of f on [a,b].\text{Let } f \text{ be a bounded function defined on } [a,b]. \\ \ \\ \text{Then if } L(f) = U(f), \\ \text{we say that } f \text{ is Riemann integrable on } [a,b] \text{ and that} \\ \ \\ \int_a^b f = \int_a^b f(x) dx = L(f) = U(f) \\ \ \\ \text{is the Riemann integral of } f \text{ on } [a,b].
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