I have misunderstood your original comment. You are indeed correct. Edit: I just noticed that the function which maps x to 0 satisfies this condition as well,

I have misunderstood your original comment. You are indeed correct. Edit: I just noticed that the function which maps x to 0 satisfies this condition as well, but isn't the exponential function if we consider 0 as part of the domain definition. It doesn't matter much for our discussion though.