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.
- idX: (X, τ2) → (X, τ1)
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) .