If a function is defined on R with right and left hand limits

>>8576028
no

>>8576037
Proof? Geniunely curious.

Well thats a shame

>>8576040
It is the definition of continuity.

>>8576040
Continuity is defined as:
1) At every point, the limit as x approaches that point exists
2) At every point, f is defined.
3) At every point, the limit as x approaches the point must be equal to f valued at that point

Now you say that the function is defined for all the real numbers and the limit exist everywhere as well. Well, this second statement is what gives the proof. If a function has a limit everywhere then the set of points in where f is discontinuous can be, at most, countable.

The set of real numbers and any interval of real numbers is an uncountable set. So you could never construct a function like this. At best you could make it discontinuous at infinitely many points, but not all points.