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.

Question

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.

procjenomuj · Accepted Answer

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  &amp;lt;  x  &amp;lt;  2.

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