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Continuity

# 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.

Why is this function not continuous at x=-2?

This function is not continuous at x=-2 because f(-2) is not defined.

Why is this function not continuous at x=1?

• When $x\to\text{1^-}$, $\lim_{x\to\text{1^-}}\text{f(x)}=3$

• When $x\to\text{1^+}$, $\lim_{x\to\text{1^+}}\text{f(x)}=-2$

• Because $\lim_{x\to\text{1^-}}\text{f(x)}\;\neq\;\lim_{x\to\text{1^+}}$,   $\lim_{x\to\text{1}}\text{f(x)}$ does not exist.

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:
The fuction is is continuous is discontinuous