Real exponential function/Representation/Continuation/Remark

There is another way to introduce the exponential function ${}x\mapsto a^{x}$ to base ${}a>0$. For a natural number ${}n\neq 0$, one takes the ${}n$th product of ${}a$ with itself as definition for ${}a^{n}$. For a negative integer ${}x$, one sets ${}a^{x}:=(a^{-x})^{-1}$. For a positive rational number ${}x=r/s$, one sets

${}a^{x}:={\sqrt[{s}]{a^{r}}}\,,$

where one has to show that this is independent of the chosen representation as a fraction. For a negative rational number, one takes again the inverse. For an arbitrary real number ${}x$, one takes a sequence ${}q_{n}$ of rational numbers converging to ${}x$, and defines

${}a^{x}:=\lim _{n\rightarrow \infty }a^{q_{n}}\,.$

For this, one has to show that these limits exist and that they are independent of the chosen rational sequence. For the passage from ${}\mathbb {Q}$ to ${}\mathbb {R}$, the concept of uniform continuity is crucial.