Mathematics · Ch 5 — Continuity and Differentiability

Summary

Continuity at a point: A function $f$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$ . This requires three conditions: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and both are equal.
Types of discontinuity: Removable (limit exists but not equal to $f(a)$ ), jump (left and right limits differ), and infinite (limit tends to $\pm \infty$ ).
Continuity in an interval: $f$ is continuous on $[a, b]$ if it is continuous at every point in $(a, b)$ and $\lim_{x \to a^+} f(x) = f(a)$ , $\lim_{x \to b^-} f(x) = f(b)$ .
Algebra of continuous functions: Sum, difference, product, and quotient (denominator $\neq 0$ ) of continuous functions are continuous.
Differentiability at a point: $f$ is differentiable at $x = a$ if $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists (the derivative $f'(a)$ ). Differentiability implies continuity, but not vice versa.
Derivative formulas: Standard derivatives: $\frac{d}{dx} x^n = n x^{n-1}$ , $\frac{d}{dx} \sin x = \cos x$ , $\frac{d}{dx} \cos x = -\sin x$ , $\frac{d}{dx} e^x = e^x$ , $\frac{d}{dx} \ln x = \frac{1}{x}$ .
Chain rule: For composite $f(g(x))$ , $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$ .
Implicit differentiation: Differentiate both sides of an equation in $x$ and $y$ w.r.t. $x$ , treating $y$ as a function of $x$ , then solve for $\frac{dy}{dx}$ .
Logarithmic differentiation: Take $\ln$ of both sides of $y = f(x)^{g(x)}$ to simplify differentiation of products, quotients, or powers.
Derivatives of inverse trigonometric functions: $\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}}$ , $\frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2}$ , $\frac{d}{dx} \sec^{-1} x = \frac{1}{|x|\sqrt{x^2-1}}$ , etc. …

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