Reasoning on Efficiency (3)
The answer depends on your computation model. Let's consider the case of Turing machines, and the language $L$ over $\{0,1\}$ of words ending in $1$. This language has time complexity $n+1$.

Blum's speedup theorem states that optimal algorithms don't exist for all problems, in a very strong sense (which is explained in the Wikipedia article).

Finally, let me mention that your notion of optimality is too restrictive. Given an algorithm with running time $T(n)$ for any problem, it is easy to construct (in the Turing machine model) another algorithm whose running time is also $T(n)$. So no language can have an optimal algorithm in this very strong sense. Even if you consider only running times, it is easy to construct (in the same model) another algorithm with the different running time $T(n)+1$. So no non-trivial language can have an optimal running time in your very strong sense.