Let h: be continuous on satisfying =0 for all m within integers and n within naturals. Show that h(x)=0 for all x within reals
Let $\displaystyle r\in \mathbb{R}$ with $\displaystyle h(r) > 0$ WLOG. Take $\displaystyle \epsilon > 0$ so that $\displaystyle h(r) - \epsilon > 0$. Then it means there is $\displaystyle \delta > 0$ such that if $\displaystyle x\in (r-\delta,r+\delta) \implies h(x) > h(r) - \epsilon > 0$. The problem is that on any interval we can find a number having the form $\displaystyle \tfrac{m}{2^n}$. And $\displaystyle h( \tfrac{m}{2^n} ) = 0 \not > 0$. This is a contradiction.
If $\displaystyle h(r) < 0$ choose $\displaystyle \epsilon > 0$ so that $\displaystyle h(r) + \epsilon < 0$. Then there is $\displaystyle \delta > 0$ so that $\displaystyle x\in (r - \delta,r+\delta) \implies h(x) < h(r) + \epsilon < 0$. But since we can find $\displaystyle \tfrac{n}{2^m}$ in this interval it would mean $\displaystyle 0=h(\tfrac{n}{2^m}) < 0$. A contradiction.
Therefore, the only possibility is that $\displaystyle h(r) = 0$ for all $\displaystyle r\in \mathbb{R}$.