Let $f(x)$ be a piecewise function defined as $f(x) = \begin{cases} x^2 - 1 &\text{for } x \le 2 \ 3x - 3 &\text{for } x > 2 \end{cases}$. Is $f$ differentiable at $x=2$?

Core Concept

Differentiability requires the slope of the function to be the same from both sides. Because the derivative of $x^2 - 1$ at $x=2$ is $4$, and the derivative of $3x - 3$ is $3$, the function has a corner and is not differentiable.