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Functions

Continuity

A continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.

Mathematically, function f is said to be continuous at point x = a if:

1. f(a) is defined, so that f(a) is in the domain of f.

2.

exists for x in the domain of f.

3.

Discontinuities:
If a function is not continuous at a point in its domain, one says that it has a discontinuity there. There are different types of discontinuities:

Removable discontinuity.If f(a) and are defined, but not equal.
Jump discontinuity or step discontinuity, if the one-sided limit from the positive direction and the one-side limit from the negative direction are defined, but not equal.
Essential discontinuity, One or both of the one-sided limits does not exist or is infinite.

Estudy the continuity on R of the function: