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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.

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    $\begingroup$ 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. $\endgroup$
    – Nathaniel
    Nov 22 at 20:48
  • $\begingroup$ 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 $\endgroup$
    – D.W.
    Nov 22 at 22:02

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