Is the linear map T: C^1 (R,R) --> C^0 (R, R), T(f) = f ' - f

an isomorphism?

I can't seem to find counterexamples for both injectivity and surjectivity. The kernel contains the 0 function as well as the exponential function, but I don't think this necessarily means that it isn't an isomorphism since T isn't a self-map.