I think it isn't, seeing as the left hand limit for lim(x->0) (f(x)-f(0))/x and the RHL for x->1 do not exist.

First of all you should clarify the definition of a function differentiable or, for that matter, continuous on a closed interval or, perhaps, any other subset of the reals.

I would assume that a meaningful defintion only relates to the values of x from the given set. Thus, since the limit from the left involves points not in the set, this consideration is irrelevant to the characterization of the given function.

The function at hand can be extended to, say, interval <-1,1> in a variety of ways. For some it won't even be continuous at 0, for others it will be continuous but not differentiable. It may also be made differentiable at 0 in many ways.