At what values ​of "a" is the given function continuous on the whole number axis?

Question

Question:

At what values ​​of $a$ is the function $f (x)=\dfrac{x+a-2}{a-x^2}$ continuous on the whole number axis?

My attempts:

$$a-x^2≠0 \Longrightarrow a≠x^2≥0 \Longrightarrow a<0 \Longrightarrow a\in(-\infty,0)$$

Here, I accepted the whole number axis is equivalent to $a\in\mathbb R.$

Is my solution correct?

Answer 1

You have arrived at the correct solution, but for a complete written solution (assuming that $a\in \mathbb{R}$ and that the number axis refers to the real number line), I think an additional look into what happens if $$x+a-2\mid a-x^2$$ is worth a check and go. For instance if $a=1$, $f(x)=\dfrac{-1}{1+x}$ and if $a=4$, $f(x)=\dfrac1{2-x}$, both of which are of course, eliminated from our consideration by their one-point discontinuities.

Just a matter of rigour, if you want.