# 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?

I have a rational function $f\left(x\right)=1/\left({x}^{2}-4\right)$. 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 $-\mathrm{\infty }$ 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.
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procjenomuj
To be continuous on an interval [a,b], your function must (as a starting point!) be defined for every $x\in \left[a,b\right]$. 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.