How to prove $L \notin \texttt{DSPACE}(f)$
Unfortunately, when you tried to ask a general question to get hints for your homework exercise rather than the solution, you turned the question into something too general to answer. I'll try to come back at some point and write something more useful, now that you've made the question more specific. Diagonalization is the usual technique, similar to how the halting problem is proven undecidable.



You're not going to prove that a completely generic language&nbsp;$L$ is not in $\mathrm{DSPACE}(f(n))$ for a completely generic function&nbsp;$f$.  Any such proof is going to have to rely on some properties of $L$ and&nbsp;$f$: quite apart from anything else, if you fix a decidable language&nbsp;$L$, then there are infinitely many functions&nbsp;$f$ such that $L$ is in $\mathrm{DSPACE}(f)$ and, conversely, if you fix any "reasonable" function&nbsp;$f$, there are infinitely many languages in $\mathrm{DSPACE}(f)$. So you can't hope to prove $L\notin\mathrm{DSPACE}(f)$ as a generic result.

Second, saying "I don't know any languages in $\mathrm{DSPACE}$" is like saying I don't know any people who weigh kilograms (or pounds, if you prefer). It doesn't really mean anything until you specify how many kilograms or, correspondingly, what space bound you're talking about.