the function is continuous on which set

Question

Let $f$ be a function defined on the set of real number $$f(x)=\begin{cases}1 \ \ \ \text{if $x$ is rational}\\ e^x\ \ \ \text{if $x$ is irrational}\end{cases}$$
Then on which set $f$ is continuous .I think it is continuous on set of rational. Because every constant functions is continuous, also I am confused because the set of exponential function also continuous

Answer 1

Hint

As $\lim\limits_{x \to 0} e^x = 1$, $f$ is continuous at $0$.

This is the only point of continuity. The reason is that: (1) the exponential function is continuous, (2) that $e^x \neq 1$ for $x \neq 0$ and (3) that the irrational numbers are dense in the reals.