Counter examples regarding statements on limits

Question

These are questions from Wade, Introduction to Real Analysis:

The problem goes f and g are real functions defined at all points in some open interval containing a except possibly at a.

Suppose $(x_n)$ is a sequence converging to a with $x_n \not= a$. If $f(x_n) \rightarrow L$ as $n \rightarrow \infty $ then $f(x) \rightarrow L$ as $x \rightarrow a$.

My initial attempt to was to think that the statement is false because you can take x at a such that x converges to a. But since the function is undefined at a, $f(x)$ cannot possibly converge to L. But it doesn't 'feel' right somehow. And doesn't feel like a concrete counterexample. Also, I might be a little befuddled about the difference between $(x_n)$ being convergent and $x$ being convergent.

Answer 1

Not sure where $g$ comes into the picture. But think about this classic function

$$f(x) = 0 \hbox{ if } x \in \mathbb{Q}, \ \ \ \ \ f(x) = 1 \hbox{if } x \notin \mathbb{Q}$$

Let $x_n$ be the the decimal expansion of $\sqrt{2}$ to $n$ decimal places. Hence ...

Answer 2

I think you are confused because you are thinking about continuity. The function does not have to be defined at that point to converge to it. If it is a continuous function, then it must be equal to the limit at that point, which won't happen if it's undefined at that point, but it will still have a limit L if it is the case you described regardless.