Calculus AR

Derive the technique of u substitution from the chain rule.

A function $f(x)$ is continuous at a point $x=c$ iff it meets the following three conditions:

1. $f(c)$ exists (c lies in the domain of f).
2. $\lim x\to cf(x)$ exists (f has a limit as x tends towards c).
3. $\lim x\to cf(x)=f(c)$ (the limit equals the function value).

We define a continuous function as one that is continuous at every point in its domain.