Answer
Function $f$ cannot have any continuous extension to $x=0$ because the limit does not exist as $x\to0$.
Work Step by Step
$$f(x)=\sin\frac{1}{x}$$
We will need to draw the graph of the function $f$ in this exercise. The graph has been drawn and enclosed below.
- Condition for continuous extension for function $f(x)$ at a point $c$: $\lim_{x\to c}f(x)$ must exist.
Here, from the graph, we see that $f(x)$ oscillates too much as $x\to0$ for $\lim_{x\to0}f(x)$ to exist.
Therefore, function $f$ cannot have any continuous extension to $x=0$, because the limit does not exist as $x\to0$. The existence of limit is essential, because we need a value to extend the value of $f(c)$ to be equal with so that $f$ is continuous at $c$.
