How to prove or disprove that f is computable?
Define the function $f: \mathbb{N} \to \{0,1\}$ as follows.  Let $f(x) = 1$ if turing machine $x$ halts on itself as input and $f(x) = 0$ otherwise.  Then for all $x$, we have that $f(x) \leq 1$.  Yet, if $f$ is computable, then the Halting problem should be decidable.