Differentiability

Definition: A function $f$ is said to be differentiable at $x=a$ if and only if $f'(a)$ exists.

In other words, if and only if the limit $\displaystyle \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ exists, or, equivalently, the limit $\displaystyle \lim_{x \to a} \frac{f(x)-f(a)}{(x-a)}$ exists. A function $f$ is said to be differentiable on an interval $I$ if $f'(a)$ exists for every point $a \in I$.

Differentiability and Continuity

Theorem: If a function is differentiable at $a$, then it is also continuous at $a$.

However, it can be continuous without being differentiable!

How can a function fail to be differentiable?

There are several ways that a function can fail to be differentiable. In particular:

The function may have a discontinuity. The function may have a change in direction, e.g., $f(x) = |x|$ at $x=0$. The function may have a vertical tangent, e.g., $f(x) = x^{1/3}$ at $x=0$.