# 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

**Exercise**: study the continuity of these functions and classify their discontinuities if they have them:

Solution: a) f is continuous in **R**-{0,1}, in x = 0 f has an essential discontinuity and in x = 1 f has an infinity jump discontinuity

b) f is continuous in R-{-1,1}, in x = -1 f has a jump discontinuity, with jump 3, and in x = 1 f has a removable discontinuity

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