real analysis - function continuous and differentiable at exactly one point

Is it possible for there to be a function $f:{\Bbb R} \rightarrow {\Bbb R}$ defined on some interval $(-\epsilon, \epsilon)$ for some $\epsilon \gt 0$ which is:

• continuous only at 0


• differentiable only at 0

If so, what would be the criteria for this? Or, is it required that for the derivative to exist at a point, the function must be continuous on some positive length interval containing that point?

Edit: Is there example of such function that doesn't make use of sets like "the rationals" or "the irrationals" in its definition?

Edit 2: What about some expression in x that only makes use of +,-,*,/, exponentiation, limits and infinite sequences. And, let's say you generate a sequence of such functions $f_1(x), f_2(x), ...$. Can you find such a sequence of functions such that the limiting function is continuous and differentiable only at zero?