Differentiable but Derivative not Continuous

This is the graph of the function. Zoom in by clicking and draging the mouse, or select Options to change the the power of x. Experiment with different powers. Is the function still differentiable at zero if n = 1 or n = 0 ? If yes, is the derivative continuous ? How about if n > 2 ?

Your browser can not handle Java Applets. Get latest version of Netscape now !

Proof:

We have studied this function before. The details are left as an exercise, but here is the idea:

• g is differentiable for all non-zero x (apply some theorems)
• find the derivative for non-zero x by using chain and product rules
• find the derivative of g at zero by looking at the limit of the difference quotient at zero
• show that the derivative you have just found is not continuous at zero by considering left and right limits as x approaches zero