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If a function has a limit from the right but not from the left, is it still continuous?

Rodrigo Stv
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3 Answers3

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It is continuous at $0$.

By construction, the domain of the square-root function is $\mathbb R_+=[0,\infty)$. Now, for any sequence $(x_n)_{n\in\mathbb N}$ in the domain (that is, $x_n\geq 0$ for all $n\in\mathbb N$) that converges to $0$, one has that the corresponding function values $\sqrt{x_n}$ also converge to $\sqrt{0}=0$.

And this is all you need for continuity by (one of the multiple equivalent) definition(s) of this concept in general metric spaces. What goes on “from the left” is outside of the domain and hence outside of interest as far as continuity is concerned.

triple_sec
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    Thanks for this. It must always be stressed that continuity can be considered only for points in the domain of the function. Failure to do this results in the (false) statement that $f(x)=1/x$ is discontinuous at $0$, a common error of many texts. – Lubin Jan 23 '19 at 16:32
  • @Lubin in what sense 1/x is discontinuous is a false statement? Usually when saying this, textbooks assume the so called infinity type of discontinuity, which apply precisely to points where a function is not defined and tends to infinity. I do understand 1/x is continuous on (0,infty) if you mean that, but I wouldn’t say it is false to say that as a function on R it has an infinity type discontinuity at zero. – xyz Apr 26 '21 at 00:56
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    Golly @GGG , what is “infinity”? Is the reciprocal function defined at $0$ or not? If it is defined, then I guess you have to specify what the value is. That one value would have to be some “$\infty$” creature, and then you’d need to specify what its neighborhoods were. Once you do all of that, then I’ll talk to you about the function being defined but discontinuous at zero. – Lubin Apr 26 '21 at 18:49
  • @Lubin Pardon but isn t that exactly what the extended real line is for? – xyz Apr 27 '21 at 16:33
  • Well, @GGG , which extended real line do you choose? One where there is an infinity at each end, or one with a single infinity? If the former, you have to decide which infinity you choose for a value at zero, in which case the reciprocal function certainly is discontinuous. If a single infinity, you’re speaking, essentially, of the projective line, and the reciprocal function is continuous. Too many choices, for my money. – Lubin Apr 27 '21 at 19:42
  • @Lubin I understand, but in real analysis it is just a very common choice to work directly in R extended with the two infinities rather than with the compactification at one infinity, at least at elementary level. I’d say I ve never come upon a course that does not say 1/x is discontinuous at zero. And as I mentioned it is largely a matter of definitions and taste, but at the base there is the idea of working in this topological setting, more or less formally depending on the audience. Regardless, it is very very common and accepted, so I was simply surprised to hear it was false tout court. – xyz Apr 27 '21 at 20:34
  • @GGG: See discussion here: https://math.stackexchange.com/questions/1087623/is-function-f-mathbb-c-0-rightarrow-mathbb-c-prescribed-by-z-rightarrow – Hans Lundmark Sep 12 '21 at 14:18
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One can only talk about continuity on the domain where the function exists. The function doesn't exist for $x < 0$ so we can't talk about limits for values $x < 0$.

$\sqrt x $ is continuous everywhere it exists.

fleablood
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By definition if the approaches differ, or if one doesn't exist, the limit is undefined, so the function can't be continuous. In this case, however, the limit does exist if you treat it as a function from the complex numbers to the complex numbers, with it approaching zero in the neighborhood of the origin.

user361424
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