1. ## Differentiable piecewise function

Let $\displaystyle f(x) = x^2 \text{for } x \geq 0 \wedge f(x) = 0 \text{for } x < 0.$
B. Show that f is differentiable at $\displaystyle x = 0.$
C. Calculate $\displaystyle f' \text{on } \mathbb{R}.$

Any help will be greatly appreciated.

-the Doctor

2. Originally Posted by thedoctor818
Let $\displaystyle f(x) = x^2 \text{for } x \geq 0 \wedge f(x) = 0 \text{for } x < 0.$
B. Show that f is differentiable at $\displaystyle x = 0.$
C. Calculate $\displaystyle f' \text{on } \mathbb{R}.$
I don't understand. It's clearly differentiable on $\displaystyle \mathbb{R}-\{0\}$ and $\displaystyle \lim_{x\to0^-}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^-}0=0=\lim_{x\to0^+}2x=\lim_{x\to0^+}\frac{f(x)-f(0)}{x-0}$
3. Since the $\displaystyle lim_{x\to0-}=lim_{x\to0+}$ then does that mean that $\displaystyle f(x) \; \text{is differentiable at } x=0$?