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Consider a real valued function $f$ defined on $[a,b]$. Say it has a simple discontinuity at a point $x \in (a,b)$, with $f(x-) \neq f(x+)$ (LHL not equal to RHL).

Is it necessary that $f(x)$ is equal to either $f(x-)$ or $f(x+)$, or can it be different from both?

Kindly help!

bitter-sweet
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  • What do you mean by a simple discontinuity? – Tobias Kildetoft May 24 '16 at 12:38
  • @TobiasKildetoft Oh sorry, that is Walter Rudin's terminology. It's a discontinuity of the "first kind", where both LHL and RHL exist. Either they aren't equal to each other, or they are without being equal to the value of the function at that point. – bitter-sweet May 24 '16 at 12:51
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    In that case you can just define the value of the function to be any value not equal to one of the limits. – Tobias Kildetoft May 24 '16 at 12:53

1 Answers1

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Consider the function $f(x) = 0$ when $x < 2$;

$f(x) = 1$ when $x = 2$;

$f(x) = 2$ when $x > 2$;

Here, $f (2-) = 0;$ $f (2+) = 2;$ $f (2) = 1$

MRobinson
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