Show that if f is strictly increasing on S and f(S) is open, then f is continuous on S

Question

Here is the problem from Mathematical Analysis by T. Apstol.

And this is the solution of part (a),

I don't understand the highlighted part. There's no assumption of continuity, but how he could he know that y1=f(x1) for some x1? Is it because the ball of f(a) is contained in f(S), so around f(a) is the image of S, so it's allowed to call y1 in B(f(a), e)as f(x1)?

If possible, could you give alternative proof or hint?