Suppose $\displaystyle f(x)$ is a differentiable function whose derivative is $\displaystyle \dfrac{1}{1+x^2}$. We know that $\displaystyle f(x) = \arctan{x}+C$. Now, suppose $\displaystyle g(t) = \tan(t^3+4t^2-2t+7)$ and $\displaystyle h(t) = \cot(t^3-7t^2-34t+1)$. Suppose we want to figure out what $\displaystyle f(h(t))-f(g(t))$ is. Taking the derivative, we get $\displaystyle f^\prime(h(t))h^\prime(t)-f^\prime(g(t))g^\prime(t) = 6(t+2)(t-3)$. Integrating, we get $\displaystyle 2t^3-3t^2-36t+C$ which undoubtedly achieves values outside the range of $\displaystyle \arctan(x)$ regardless of the constants for each. Is this because $\displaystyle \tan(x)$ is discontinuous?