# We have function f(x) and we know it has point x=a, which is a point of discontinuity(jump)

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.

• Short version: yes you are! Do you have any uncertainties? – Milloupe Nov 26 '19 at 10:40
• Thank you for feedback. :) – naruto25 Nov 26 '19 at 11:02

Yes, you are right. $$sgn(x)$$ is a counter example to first three of them.