Existence of the inverse of a function

Question

Let $a \in \mathbb{R}$ and $a>0 \wedge a\neq 1$. We define $f: \mathbb{R} \rightarrow \mathbb{R}_{>0}$ through $f(x)=a^x$. I read in a german book that f's monotone increase/decrease for $a>1 / 0<a<1$ and its continuity imply the existence of the inverse function. Does that hold? Since today I thought we need the stronger case of $x<y \Rightarrow f(x)<f(y) \text{or} f(x)>f(y)$.

Answer 1

Hint: $f(x)=e^{x \ln a}=e^{cx}$ with $c= \ln a.$

Hence $f'(x)=c e^{cx}$.

If $a>1$, then $c >0.$

If $0<a<1$, then $c <0.$

Can you proceed ?