Assume that function $f$ is continuous at $x=0$. Prove that the function $f(x)=a^x$ for $a>0 $, is continuous at every real number.
I know that $f$ is continuous at 0 if and only if 0 is in the domain of $f$ and $lim_(h→0)〖{f(0+h)-f(0) ]=0〗$. But how can I use this to prove this problem?
It wants you to assume that it's continuous at $x=0$, and use that to prove that it's continuous everywhere else. In particular, you're assuming that $$\lim_{x\to 0}|a^x-1|=0.\tag{$\clubsuit$}$$ Now, note that for any $x_0$, we can write $$|a^x-a^{x_0}|=|a^{x_0}||a^{x-x_0}-1|.\tag{$\heartsuit$}$$ Using $(\clubsuit)$ and $(\heartsuit)$ together with the fact that $|a^{x_0}|>0$ for all $x_0$, we can prove continuity anywhere else with an $\epsilon$-$\delta$ style approach.
