Are we suposed to consider $f(0)=1$?

Question

If you plot the graph of $f(x)=\frac{\sin x}{x}$ on GeoGebra, you would realise that the function is shown to be continuous at $x=0$.

I do agree the $\lim_\limits{x\to 0}\frac{\sin x}{x}=1$, but $\frac{\sin0}{0}$ is a not a valid statement.

Thus, in general when we approach a question should we take $f(0)=\lim_\limits{x\to 0} f(x)$, or should we consider $f$ to be discontinuous at exactly one point (i.e. at $x=0$)?

Answer 1

$f$ is not defined at $0$, so it is neither continuous nor discontinuous at $x=0$.

However, you can also define a function

$$g(x)=\begin{cases}\frac{\sin x}{x} & x \neq 0\\ 1& x=0\end{cases}$$

which is a continuous function on all real numbers, and is equal to $f$ on the domain of $f$.

Answer 2

You can not say that this function is discontinuous at $x = 0$. It is not defined at $0$.

However, you can extend the function $f\colon \mathbb{R} \setminus \{ 0 \} \to \mathbb{R}$ given by $f(x) = \sin x / x$ setting $$ \tilde f(x) = \begin{cases} f(x), & x \neq 0,\\ 1,& x = 1. \end{cases} $$ Such $\tilde f$, defined on $\mathbb{R}$, is now continuous.