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I'm trying to understand if a function is continuous at $x=a$ if $lim_{x \rightarrow a^+}f(x)=+\infty$ and $lim_{x \rightarrow a^-}f(x)=+\infty$? Now, I know that if the left hand limits and right hand limits are equal then the function is continuous. However, since the "limit" here is $+\infty$ I'm wondering if the function would still be continuous since it wouldn't actually be defined at $x=a$.

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The left hand side and right hand side limit are equal to each other in the sense that they are both +infinity. But for continuity, the limits should be finite. So in your case, $y=\frac{1}{x^2}$ would serve as an example, but this function is clearly not continuous at $x=0$

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  • $\begingroup$ So if the left and right limits are not finite, regardless if whether they are equal to each other or not, then the function can't be continuous at that point? $\endgroup$ – Hai Oct 12 '17 at 20:34
  • $\begingroup$ If the limits are not finite, then presumably they are infinite. Graphically this is commonly orchestrated by a vertical asymptote. At that $x$ value, the function is discontinuous (and your $x=a$ is not in the domain of the function) $\endgroup$ – imranfat Oct 12 '17 at 20:36
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    $\begingroup$ For a function to be continuous at $x=a$, $f(a)$ must exist. $\endgroup$ – Bernard Massé Oct 12 '17 at 20:46

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