Did I make a mistake when finding the intervals this function is continuous on?

Question

I was given a function $f$ given by$$f(x) = \begin{cases} \dfrac{x^2 - a^2}{x - a \;}&\text{if} \; x \neq a, \\ 2a & \text{if} \; x=a, \end{cases}$$ and told to find the intervals over which $f$ is continuous, in terms of $a$. I just figured it is continous everywhere except $a$, so the intervals would be $(-\infty, a)$ and $(a, \infty)$. However, my textbook gives the answer, without working, as $(-\infty, -a)$ and $(-a, \infty)$. I'm like $99\%$ sure this is a mistake by the textbook author, not me, but continuity and the like is not my strong side, so I just want to make sure I didn't make a mistake here. Thanks in advance!

Answer 1

It is $$\frac{x^2-a^2}{x-a}=x+a$$ if $x\neq$ $a$ and $$\lim_{x\to a}\frac{x^2-a^2}{x-a}=2a$$ thus our function is continous for all real $x$