I learned the following theorem in class: "If $T(n)$ is asymptotically non-decreasing and $f(n)$ is smooth, then $T(n) = O(f(n)|n=b^k, k=integer)$ implies $T(n) = O(f(n))$."
I'm trying to show using a counterexample that this theorem doesn't hold if $f(n)$ is non-decreasing but not smooth. Smoothness means that $f(bn) = O(f(n))$.
From my understanding, the above conditions mean that $f(n)$ must be an exponential. Other than that, how would I go about finding a function that meets the conditions outlined above?
Since this is a homework problem, please don't give an actual function, but instead give a series of steps to take to find the function. Thanks.