# Continuity

In mathematics, 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**.Continuous function is a function whose graph can be drawn without lifting the pen from the paper.

# Definition of continuity of a function

A function

*f*(*x*) is said to be continuous at*x*=*a*, if = =*f*(*a*), i.e., LHL = RHL = value of a function at*x*=*a*or =*f*(*a*).A function

*f*(*x*) is said to be discontinuous at*x*=*a*, if- and exist but are not equal.
- and exist and are equal but not equal to
*f*(*a*). *f*(*a*) is not defined.- At least one of the limits does not exist.

**of a function is the property of interval and is meaningful at**

*Note:**x*=

*a*only if the function has a graph in the immediate neighborhood of

*x*= a, not necessarily at

*x*=

*a*. Hence it should not be mislead that continuity of a function is talked only in its domain.

For example, continuity of

*f*(*x*) = at*x*= 1 is meaningful but continuity of*f*(*x*) = log_{e}*x*at*x*= –2 is meaningless. Similarly, if*f*(*x*) has a graph as shown then continuity at*x*= 0 is meaningless.Also continuity at

*x*=

*a*⇒ existence of limit at

*x*=

*a*but existence of limit at

*x*=

*a*does not mean continuity at

*x*=

*a*.

# Directional continuity

A function may happen to be continuous in only one direction, either from the “left” or from the “right”.

A

**right-continuous**function is a function which is continuous at all points when approached from the positive infinity.Likewise a

**left-continuous**function is a function which is continuous at all points when approached from the negative infinity.# Continuity in interval

A function is said to be continuous in the open interval (

*a*,*b*) if*f*(*x*) is continuous at each and every point ∈ (*a*,*b*). For any*c*∈ (*a*,*b*) = =*f*(*c*).A function

*f*(*x*) is said to be continuous in the closed interval [*a*,*b*] if it is continuous at every point in this interval and the continuity at the end points is defined as*f*(*x*) is continuous at*x*=*a*if*f*(*a*) = = RHL (LHL should not be evaluated) and at*x*=*b*if*f*(*b*) = = LHL (RHL should not be evaluated).