TheoremsDefinitionsProof MethodsProblem Solving Techniques

Catalog of proof methods and techniques with examples.

Proof Method 1. Prove Continuity

Use the ε-δ definition of continuity at a point to prove that a given function is continuous at a given point.

Examples:
Example 1:
Let f:D→R and let c be an isolated point in D.Show that f is continuous at c.\text{Let } f: D \to \R \text{ and let } c \text{ be an isolated point in } D. \\ \text{Show that } f \text{ is continuous at } c.
Example 2:
Define f:R→R by f(x)=x2+3x−5.Use Definition 4‾ to prove that f is continuous at 3.\text{Define } f: \R \to \R \text{ by } f(x) = x^2 + 3x - 5. \\ \text{Use \underline{\href{/notes/intermediate-analysis/definitions#definition-4}{\text{Definition 4}}} to prove that } f \text{ is continuous at } 3.
Example 3:
Define f:R→R viaf(x)={xsin⁡1xx≠00x=0.Show that f is continuous at x=0.\text{Define } f: \R \to \R \text{ via} \\ f(x) = \begin{cases} x \sin \frac{1}{x} & x \neq 0 \\ 0 & x = 0. \end{cases} \\ \text{Show that } f \text{ is continuous at } x = 0.
Proof Method 2. Prove Discontinuity

Use the ε-δ definition of continuity at a point to prove that a given function is discontinuous at a given point.

Examples:
Example 1:
Define f:R→R by f(x)={5xx∈Qx2+6x∈R∖Q.Prove that f is discontinuous at 1.\text{Define } f: \R \to \R \text{ by } f(x) = \begin{cases} 5x & x \in \mathbb{Q} \\ x^2 + 6 & x \in \R \setminus \mathbb{Q}. \end{cases} \\ \text{Prove that } f \text{ is discontinuous at } 1.
Example 2:
f(x)={0if x<01if x≥0 is discontinuous at x=0.f(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases} \text{ is discontinuous at } x = 0.
Proof Method 3. Prove Continuity Using Sequential Condition

Use sequences (Theorem 5.2.3(b)) to prove that a given function is continuous at a given point.

Examples:
Example 1:
Let f(x)=x2. Show that f is continuous at 2.\text{Let } f(x) = x^2. \text{ Show that } f \text{ is continuous at 2}.
Proof Method 4. Prove Discontinuity Using Sequential Condition

Use sequences (Theorem 5.2.7) to prove that a given function is discontinuous at a given point.

Examples:
Example 1:
Let D=(−∞,0)∪(0,∞) and let f(x)=1x  ∀x∈D.Show that f is continuous on D, but not on R.\text{Let } D = (-\infty, 0) \cup (0, \infty) \text{ and let } f(x) = \frac{1}{x} \ \ \forall x \in D. \\ \text{Show that } f \text{ is continuous on } D, \text{ but not on } \R.
Proof Method 5. Prove Uniform Continuity

Use the ε-δ definition of uniform continuity to prove that a given function is uniformly continuous on a given domain.

Examples:
Example 1:
Prove that f(x)=x3 is UC on [0,2]\text{Prove that } f(x) = x^3 \text{ is UC on } [0,2]
Example 2:
Prove that f(x)=1x is UC on [2,∞)\text{Prove that } f(x) = \frac{1}{x} \text{ is UC on } [2,\infty)
Example 3:
Prove that f(x)=x is UC on [0,∞)\text{Prove that } f(x) = \sqrt{x} \text{ is UC on } [0,\infty)
Proof Method 6. Disprove Uniform Continuity

Use the ε-δ definition of uniform continuity to prove that a given function is not uniformly continuous on a given domain.

Examples:
Example 1:
Show that f(x)=x2 is not UC.\text{Show that } f(x) = x^2 \text{ is not } UC.
Proof Method 7. Prove Differentiability

Use the definition of the derivative to show that a given function is (or is not) differentiable at a given point. There are several possibilities here: one can use one-sided limits, one can use the sequential criterion for limits (Theorem 5.1.9), or one can use the ε-δ definition of limit of a function at an accumulation point.

Examples:
Example 1:
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Proof Method 8. Prove Integrability

Use the definition of integrable function to prove that a given function is integrable.

Examples:
Example 1:
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