Question

Prove that the exponential function, $a^x$ is continuous at every point ( where $a>0$ )

Answer 1

We have from the limit property, if a sequence $\left\{x_n\right\}$ converging to $c$ then for $a>0$, $\left\{a^{x_n}\right\}$ also converges to $a^c$.......(1).

Now we will prove the continuity of $a^x$ at any arbitrary point $c$ by sequential criteria.

[Sequential criteria for continuity of a function $f\left(x\right)$ at $x=c$ says,

The function $f\left(x\right)$ is said to be continuous at $x=c$ if and only if, for every sequence $\left\{x_n\right\}$ converging to $c$ gives the sequence $\left\{f\right(x_n\left)\right\}$ also converges to $f\left(c\right)$.]

Let $f\left(x\right)=a^x$ and also let sequence $\left\{x_n\right\}$ be any arbitrary sequence converging to $c$,

The using (1) it is clear that $\left\{f\right(x_n\left)\right\}$ converges to $f\left(c\right)$.

Then by sequential criteria we have $f\left(x\right)=e^x$ is continuous at $x=c$.

Since $c$ being any arbitrary point, then $f\left(x\right)$ is continuous at every pont $\in R$.