Question: If the product is defined as 
$$(f,g) = \int\_{-\infty}^\infty f'(x)\overline{g'(x)} \, dx$$

then is the space of continuously differentiable functions with finite support an inner product space?

I initially thought No because if $f$ is the Cantor function then $(f,f)=0$ however $f\neq 0 $. While $f$ is differentiable almost everywhere and is continuous everywhere.... maybe it is not $C^1$ because we require more than just a.e. continuous differentiability to be in $C^1$?

Answer: 
We require everywhere differentiability, with a *continuous differentiate* for functions in $C^1$.

But you're on the right track, exactly the *constant* functions will have $(f,f)=0$ with respect to this positive semidefinite bilinear form.
It is an inner product space, but it is not complete ( so, not a Hilbert space).