# How do we prove the following regarding the worst case running time?

Consider a randomly chosen function $$f(n)$$ over positive integers ($$n$$) such that $$n \leq ceil(f(n))$$. How do we prove that:

For any such chosen $$f(n)$$: There always exists a problem $$P$$ such that: The most efficient algorithm for $$P$$ has the worst case running time of $$f(n)$$.

We are assuming a standard Universal Turing Machine model as a reference/foundation.

New contributor
xyz is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• I did not downvote, but I assume that it is because you juste stated a problem without context and expected the community to solve it for you without sharing your thoughts or explaining your difficulties in solving it. Nov 22 at 20:48
• Please edit the question to explain the context: where did you see this claim? why do you think it is true? what is the motivation? I am skeptical whether it is true. See, for example, en.wikipedia.org/wiki/Constructible_function, math.stackexchange.com/q/51082/14578
– D.W.
Nov 22 at 22:02