Discontinuities

There are different kinds of discontinuities:

–Removable discontinuity: if the limit exists and it is not equal to f(a).

Example 1:   

      f has a removable discontinuity in x = 1

–Jump discontinuity: when the lateral limits exist and they are not equal. The jump can be finite or infinite.

Example 2:

   f has a jump discontinuity in x = 1, with jump 1.

Example 3:

  f has an infinity jump discontinuity in x = 0

  Example 4: 

    does not exist 

Exercises:

1.- Study the continuity of these functions and classify their discontinuities if they have them:

2.- Study the continuity of this function depending on the parameter a:

Solutions:

1.- a) f is continuous in R-{-1}, in x = -1 f has a jump discontinuity with jump (e - 2); b) g is continuous in R

2.- if a = -1, f is continuous in R

   - if a ≠ -1, f is continuous in R-{2}, in x = 2 f has a jump discontinuity with jump (3 + 3a)