# 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

**Essential discontinuity**: when one of the lateral limits does not exist.

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 + 3*a*)

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