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?

Denisse Fitzpatrick 2022-09-30 Answered
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 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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Answers (1)

procjenomuj
Answered 2022-10-01 Author has 8 answers
To be continuous on an interval [a,b], your function must (as a starting point!) be defined for every x [ 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.
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