Define f(x =x2 if x < 0 and f(x =x x2 if x> =0. differentiation gives f''(x2. this is bogus. why

In order for to be differentiable at , i.e. for to exist, the limit from either side of of must be the same. This is not the case, as




and so does not exist.

Meanwhile,



where



But we found that doesn't exist, so this limit also can't exist, which in turn means that does not exist.

On the other hand, is continuous and differentiable everywhere, so that , and in particular, .