TITLE: Is there a function which approaches the same value from both sides but is defined differently at that point?
QUESTION [2 upvotes]: e.g. a function, f(x) where f(0) approaches 0 from positive x and 0 from negative x, but when you actually compute f(0) it is defined as a different value like 1? And I don't mean some function where you define it differently for x > 0, x < 0 and x = 0.
Additionally a function where the limit at an x approaches a different number from positive x than from negative x, but is actually defined at that point. When the limit approaches a different value from two different directions usually we say it is therefore undefined at that point, but is this always the case?

REPLY [0 votes]: I just thought of this one: $y=0^{|x|}$. From this you can then have $y=0^{|x|} + x^2$ and so on.
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