Does there exist a function $f:\Bbb{R}\to\Bbb{R}$ that is only differentiable at $x=0$ with $f'(0)\neq0$ ?
This is a straightforward question to ask, but I'm having difficulty proving it rigorously.
I know that it is possible for functions to be differentiable only at a single point (see Is there a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable? and Function which is continuous everywhere in its domain, but differentiable only at one point). Moreover, I know of at least one function that is differentiable only at $x=0$ with the derivative there being $0$ (see https://planetmath.org/functiondifferentiableatonlyonepoint for the example).
My question is: can you have a function that is differentiable only at a point with the derivative there being non-zero? Intuitively, the answer seems to be "NO" given that a non-zero derivative at a point implies that the function is "changing" (in a somewhat vague sense) about that point, which in turn suggests that there's a small neighborhood around the point where other derivatives would exist. Seeing that the function is only differentiable at one point, this would be a clear contradiction.
I don't know if this intuition is correct let alone how I could use it to answer the question. How should one approach proving this? Should you invoke the linearity of the function around the differentiable point? That was another idea I had.
