Define a function $f:\mathbb{R}\rightarrow \mathbb{R}$ as $f(x)=\begin{cases}\frac{\sin x}{x}& \text{if}\hspace{.2cm} x \neq 0\\ 1,& \text{otherwise} \end{cases}$. Then $f$ is

$f$ is continuous everywhere

We have $\lim_{x \to 0}\frac{sin x}{x}=1$ and $f(0)=1$. So $f$ is continuous at $x=0$. $f$ is continuous at every other point because $\sin x$ is a continuous function and $f(x)=x$ is also continuous.