Parameter for differentiability

Problem

For which value of the parameter $a$ is the function $f(x) = \begin{cases} ax & x\le 0,\\ x^2 & x>0\end{cases}$ differentiable at $x_0 = 0$?

Solution

Continuity at $0$: $f(0^-)=a\cdot 0=0$, $f(0^+)=0^2=0$, $f(0)=0$: continuous for every $a$.

One-sided derivatives: $f'_-(0) = \lim_{x\to 0^-}\frac{ax-0}{x} = a, \qquad f'_+(0) = \lim_{x\to 0^+}\frac{x^2-0}{x} = \lim_{x\to 0^+} x = 0.$ The function is differentiable at $0$ if and only if $f'_-(0)=f'_+(0)$, that is $a = 0$. Hence $\boxed{a=0}$.

Links

Topics: Derivatives
Concepts: Continuity · One-sided derivative
Skills: Reason by cases · Solve equations
Exercise type: Derivative computation