The derivative of

So by the product rule we have

I'm sorry, I was focused on the wrong thing.

The function is defined everywhere, is it not? So as you are right in saying that osillates , but the fuction has been defined to be 0 when x=0, therefore the function ossilates from the left and the right, but when x reaches 0 , and therefore, . Further more, .

Does that help at all?