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I have function f(x) and only know it has point x=a, which is a point of discontinuity(jump). Based on this information we can state:

There are 4 statements

  • f(x) cannot have domain all Real numbers
  • f(x) does not have right and left limit in point x=a
  • f(x) does have in point x=a local minimum
  • f(x) does not have limit in point x=a

Well, in my opinion, first statement is false, because sign(x) has point of discontinuity(jump) and does have all Real numbers as domain. Second is also false, because one sided limit exists situated to right/left side from "jump". Third is also false because it could be also local maximum since the jump is not specified. And the fourth is right, because when there is a jump, right and left limits are not equal and thus limit does not exist.

Am I right? Thx a lot.

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    $\begingroup$ Short version: yes you are! Do you have any uncertainties? $\endgroup$ – Milloupe Nov 26 '19 at 10:40
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    $\begingroup$ Thank you for feedback. :) $\endgroup$ – naruto25 Nov 26 '19 at 11:02
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Yes, you are right. $sgn(x)$ is a counter example to first three of them.

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