Let $\{f_k\}$ be a sequence of continuous functions $f_k:\mathbb{R} \mapsto [0,\infty).$ Which of those statements can be true? (Not simultaneously)
1) The sequence $\{f_k\}$ is not bounded iff $x$ is in $\mathbb{Q}$
2) The sequence $\{f_k\}$ is not bounded iff $x$ is not in $\mathbb{Q}$
3) $\lim_{k \to \infty} f_k (x)= \infty$ iff $x$ is not in $\mathbb{Q}$
Hint: use Baire theorem. I have no clue how to approach this problem.
To use Baire category the first challenge is to express the relevant properties in terms of countable intersections of open sets, or countable unions of closed sets. Try to write the set of points $\{x\vert f_k(x)\text{ unbounded}\}$ as a countable intersection of open sets, then note they must be dense because they contain $\mathbb Q.$ This shows that condition 1 cannot hold.
2 and 3 can hold, you just need to come up with an appropriate construction. I suggest finding a pointwise increasing sequence of piecewise linear functions such that each rational $p/q$ is eventually sent to $q$ for large enough $k$, but the value at each irrational is unbounded.