If f(x) fails to be continuous because
lim  f(x) 
x > a 
fails to exist (or is infinite), then there is no way to remove the discontinuity  the limit statement takes into consideration all of the infinitely many values of f(x) sufficiently close to a and changing a value or two will not help. We call this an essential discontinuity.
If f(x) is discontinuous at x=a because

lim  f(x) 
x > a 
fails to exist or be finite. The possibilities are:
lim  sin(/x) 
x > 0 
Thus the function
{  1  if u > 1,  
s(u) = u/u =  undefined  if u = 0,  
1  if u < 1. 
(introduced by Andron's Uncle Smith) has a jump discontinuity at u=0.
As other examples, the functions h(t) and j(t) from "Left and Righthand Limits" in Stage 3 have jump discontinuities.
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William A. Bogley
Robby Robson