Continuity and Differentiability — Formula Sheet

For the limit to exist, the Left Hand Limit (LHL) must equal the Right Hand Limit (RHL): lim(x→c⁻) f(x) = lim(x→c⁺) f(x).

f/g is continuous at x = c provided g(c) ≠ 0.

The derivative of f at c is

f'(c) = lim(h→0) [f(c+h) - f(c)] / h, provided this limit exists.

LHD at c = lim(h→0⁻) [f(c+h) - f(c)] / h

Chain Rule

If y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx).

For three levels

if y = f(v), v = g(u), u = h(x), then dy/dx = (dy/dv)·(dv/du)·(du/dx).

A function f is continuous at point c if three conditions are satisfied: (1) f(c) is defined, (2) lim(x→c) f(x) exists, and (3) lim(x→c) f(x) = f(c).

A function f is differentiable at c if the derivative f'(c) = lim(h→0)[f(c+h)-f(c)]/h exists and is finite.