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A function is partially continuous if it has at most a finite number of discontinuities and furthermore the one-sided limits at each point of discontinuity exist and they are finite.

Does the part "...the one-sided limits at each point of discontinuity exist..." mean that the one-sided limits are not equal to $\infty$??

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    $\begingroup$ The part "... and they are finite" means the one-sided limits are not $\infty$. Some people include the finiteness in the existence of a limit, others don't. $\endgroup$ – Daniel Fischer Jun 4 '14 at 21:05
  • $\begingroup$ Ahaa!!! Ok!!! Thanks a lot!!! :-) $\endgroup$ – Mary Star Jun 4 '14 at 21:23
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Not necessarily, but the part "and they are finite" means the one-sided limits are not infinite.

Having an infinite limit can be stated very precisely, and there is not an accepted convention about whether $\lim_{x \to c} f(x) = \infty$ means that $f$ has a limit (and the limit is infinite) or that $f$ doesn't have a limit (because the limit is infinite).

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  • $\begingroup$ I got stuck right now... Could you explain me the difference of $$\lim_{x \to c} f(x) = \infty \text{ means that f has a limit (and the limit is infinite) }$$ and $$f \text{ doesn't have a limit (because the limit is infinite) }$$?? $\endgroup$ – Mary Star Jun 4 '14 at 21:28
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    $\begingroup$ It's a matter of convention. Some say that if a limit is infinite, then it exists, and others don't. $\endgroup$ – davidlowryduda Jun 4 '14 at 21:35
  • $\begingroup$ Ahaa!!! Ok!! Thank you!!! :-) $\endgroup$ – Mary Star Jun 4 '14 at 21:40

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