# Thread: Differentiable piecewise function

1. ## Differentiable piecewise function

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

Any help will be greatly appreciated.

-the Doctor

2. Originally Posted by thedoctor818
My task is as follows:
Let $f(x) = x^2 \text{for } x \geq 0 \wedge f(x) = 0 \text{for } x < 0.$
B. Show that f is differentiable at $x = 0.$
C. Calculate $f' \text{on } \mathbb{R}.$

Any help will be greatly appreciated.

-the Doctor
I don't understand. It's clearly differentiable on $\mathbb{R}-\{0\}$ and $\lim_{x\to0^-}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^-}0=0=\lim_{x\to0^+}2x=\lim_{x\to0^+}\frac{f(x)-f(0)}{x-0}$

3. Since the $lim_{x\to0-}=lim_{x\to0+}$ then does that mean that $f(x) \; \text{is differentiable at } x=0$?