Types of Discontinuity - Real Analysis, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Three Types of Discontinuities 

There are 3 common types of discontinuities.

1. Infinite Discontinuity
2. Removable Gap
3. Jump Discontinuity

Let a be a real number and f (x) a real function.  Then we say

1. f creates an Infinite Discontinuity at a
if any of the one or two sided limits at a is infinite.

If f (a) is defined, then we can furthermore say that f has an infinite discontinuity at a.

2. f creates a Removable Gap at a

if the two sided limit 
 exists and is finite

but f (a) is not defined or not equal to the limit.

If f (a) is defined then we can furthermore say that f has a removable gap discontinuity at a. 

3. f creates a Jump Discontinuity at a if each of the one sided limits is finite but they don’t agree.

If f (a) is defined then we can furthermore say that f has a jump discontinuity at a.

Discontinuities 

• Every Infinite Discontinuity creates a vertical asymptote.
• The functions most often encountered in calculus are continuous on their domains (coid) and create only Infinite Discontinuities or Removable Gaps outside of their domains.
• Jump Discontinuities usually occur with piecewise defined functions.  For example, the functions |x|/x and êë [x] pictured above are piecewise defined functions that create jump discontinuities.
• A coid function is not defined at any of the discontinuities it creates.
  • The function |x|/x is a coid function that creates a jump discontinuity at zero where it’s not defined.
  • The greatest integer function int(x) is not a coid function because it's defined at each of its jump discontinuities.
• · Other types of discontinuities by example.
  • Erratic: sin (1/x) is discontinuous at zero in an erratic way.
    The limit
      does not exist
    since the function is bounded yet not steadily approaching any one location
    
  • Weird: The Rational Indicator Function