Thread: Show that there exists no sequence of functions satisfying the following

1. Show that there exists no sequence of functions satisfying the following

I found this interesting exercise on a topology book I'm reading, but I don't have a clue what to do.

Show that there is no sequence {g_n} of continuous functions from R to R such that the sequence {(g_n)(x)} is bounded iff x is rational (where R = set of real numbers).

2. $\mathbb{R}$ is complete. Use the Baire's theorem.

Fernando Revilla