This animation shows that f(x) is not differentiable at the origin. It shows the secant line through (0,f(0)) and (h,f(h)). If f(x) were differentiable, these secant lines should approach a tangent line as h approaches zero. You can see in this animation that this doesn't happen; the secant line continues to osciallate between y=x and y=x. What happens when you replace x with x^{2}? 
This function is continuous at the origin; many books prove this as an application of the Squeeze Theorem. Because , it follows that . The Squeeze Theorem says Hence , so is continuous at . However, is not differentiable at . We can verify this using the limit definition of the derivative:
