The continuous functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x) = g(x)$ for all $x \in \mathbb{Q}$.

(a) Using the definition of continuity, prove that $f(x) = g(x)$ for all $x \in \mathbb{R}$. (b) Use sequential criteria of continuity to redo the problem.

I was able to do the part (b) of this problem by considering an arbitrary irrational number and considering a rational sequence converging to it and using the sequential criteria of continuity. what i'm struggling is with the first part, i've tried the usual method of using the continuity definition and finding a delta but i wasn't able to find a suitable one. Then i've thought of using the method that proved the definition of continuity by supposing the sequential criteria. But it felt wrong as the problems says to use the definition and then the sequential criteria instead of using the proof of the theorem itself.