Is $\frac{x-|x|}{x}$ continuous? It should be discontinuous at x=0 as left hand limit and right hand limit at zero are unequal. But it is continuous and the reason given is that they have excluded zero as its domain. Is it possible?
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Related: http://math.stackexchange.com/questions/1482787/can-we-talk-about-the-continuity-discontinuity-of-a-function-at-a-point-which-is/1482900#1482900 – Michael Hoppe Dec 29 '16 at 09:30
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The function is continuous everywhere where it is defined. Speaking of continuity at $0$ makes no sense since the function is undefined at $0$.
5xum
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Thank you very much. Can you clear one more doubt please? What is this concept called? Is this jump Discontinuity? – Elder Gate 1 Dec 29 '16 at 08:43
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When right hand limit and left hand limit are unequal still the function is continuous...how? – Elder Gate 1 Dec 29 '16 at 08:45
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@user402626 Well, the function is not continuous at $0$. It also isn't discontinuous at $0$. It's not defined at $0$, so talking about continuity at $0$ is like talking about what colour an invisible unicorn is. – 5xum Dec 29 '16 at 08:46
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Hint: Let $f(x)=(x-|x|)/x$ for $x \ne 0$. Then
$f(x)=0$ for $x>0$ and $f(x)=2$ for $x<0$
Fred
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