>>10878112You can’t assume that f is continuous, or that f^-1 is integrable, so your only choice is to use a proof by contradiction. Suppose f exists with the stated properties, then all values of f^-1 are a, or greater, which means they are all strictly greater than 0. The function is bijective, so f(a) is not equal to f(b). Then, the value of the integral is bounded below by a*(f(b) - f(a)), if f(a) < f(b), or the integral is bounded above by a*(f(b) - f(a)), if f(b) < f(a). In the first case, you have it that the integral must be larger than a positive number, and in the second case, you have it that the integral is smaller than a negative number, so a contradiction in both cases.
