For the function $\displaystyle f (x)=x|x|$ , show that $\displaystyle f ' (0)$ exists. What is the value?
Hmm... you should! The derivative of f at c is
$\displaystyle f'(c) = \lim\limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$.
The existence of this limit in this form requires that the one-sided limits
$\displaystyle f'(c) = \lim\limits_{x \to c-} \dfrac{f(x) - f(c)}{x - c}$ and $\displaystyle f'(c) = \lim\limits_{x \to c+} \dfrac{f(x) - f(c)}{x - c}$ exist and are equal. I think you will have to evaluate both one-sided limits and show that they are equal in order to prove that f'(0) exists in your case.