### Continuous Function and Its Properties

**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 a "discontinuous function". A continuous function with a continuous inverse function is called "bicontinuous".**

**There are many properties regarding Continuous Function . Some important Properties are as follows ;-**1) If f(x) be continuous in the closed interval [a,b] the given element , the interval can always be divided up into a finite number of sub - intervals such as | f(x') - f(x'') | < that element , where x' and x'' are any two points in the same interval .

2) A function , which is continuous in a close interval [a, b] , is bounded therein .

3) If

*f*:

*X*→

*Y*and

*g*:

*Y*→

*Z*are continuous, then so is the composition

*g*∘

*f*:

*X*→

*Z*. If

*f*:

*X*→

*Y*is continuous and

*X*is compact, then*f*(*X*) is compact.*X*is connected, then*f*(*X*) is connected.*X*is path-connected, then*f*(*X*) is path-connected.*X*is Lindelöf, then*f*(*X*) is Lindelöf.*X*is separable, then*f*(*X*) is separable.

*X*are partially ordered: a topology τ

_{1}is said to be coarser than another topology τ

_{2}(notation: τ

_{1}⊆ τ

_{2}) if every open subset with respect to τ

_{1}is also open with respect to τ

_{2}. Then, the identity map

- id
_{X}: (*X*, τ_{2}) → (*X*, τ_{1})

_{1}⊆ τ

_{2}(see also comparison of topologies). More generally, a continuous function

_{Y}is replaced by a coarser topology and/or τ

_{X}is replaced by a finer topology.

4) If the function f(x) is continuous in the closed interval [a, b] , then f(x) attains its bounds at least once in [a, b] .

5) A continuous function which has opposite signs at two points vanishes at least once between these points , that is if f(x) be continuous in the closed interval [a, b] and f(a) and f(b) have opposite signs , then there is at least one value of x between a and b for which f(x) = 0 .

6) If f(a) and f(b) are unequal and f(x) is continuous in the closed interval [a, b] , then f(x) assumes at least once every value between f(a) and f(b) .

## Comments

## Post a Comment