I have a rational function $f(x) = 1/(x^2-4)$. We know that $f(x)$ is not defined at $x = 2$ and $x = -2$ and has an infinite discontinuity at these $x$-values. However, I wanted to know if the function is continuous on the interval $(0,2]$ because we know that it is approaching $-\infty$ as $x$ approaches $2$ but if we only have the interval $(0,2]$, it is continuously going to negative infinity. So, is this function continuous in this interval or not? Thank you so much.
If a function approaches infinity at an interval, is it continuous
calculuscontinuitylimits
Best Answer
To be continuous on an interval $[a, b]$, your function must (as a starting point!) be defined for every $x \in [a, b]$. In your case, your function is not defined at $x = 2$, so it cannot be continuous on $(0, 2]$. It is, however, continuous on the open interval $0 < x < 2$.