### 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: XY and g: YZ are continuous, then so is the composition gf: XZ. If f: XY 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.
The possible topologies on a fixed set X are partially ordered: a topology τ1 is said to be coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the identity map
idX: (X, τ2) → (X, τ1)
is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function
$(X, \tau_X) \rightarrow (Y, \tau_Y)$
stays continuous if the topology τY is replaced by a coarser topology and/or τX 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) .