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Continuous Functions Theorems

Instructor: Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

In this lesson, you will learn the basics of continuous functions. Then you will look at two theorems, intermediate value theorem and extreme value theorem, pertaining to these functions.

Continuous functions

A continuous function in mathematics is defined as a function that is defined at each point in its domain. Basically, a function whose graph is an unbroken curve in its domain is a continuous function.

In mathematical terms, a function f(x) is considered to be continuous at a point c if and only if it satisfies the following conditions -


Continuous function


The functions that are continuous at every point in their domain are continuous functions.

Let's see an example.


Example Function


The domain of this function is all real numbers [- ∞, + ∞]. This function satisfies all three conditions, so it is a continuous function, as can be seen in its graph.


Graph of f(x)


Theorems of Continuous Functions

We will now look at two important theorems pertaining to continuous functions:

1. Intermediate Value Theorem

Let's assume a function f(x) that is continuous between [a,b]. Suppose there is a number p between f(a) and f(b). This theorem states that there must be at least one value q between a and b, such that f(q) = p.


Intermediate Value Theorem


Taking the example of the previous function,


Example Function


Between the range [0,4], there exists a value f(q)=4 for q=2, which can be illustrated as


X2 0 to 4


There could be functions that have more than one point between a and b with the same value. Let's consider an interval of [-4,4] for our previous example. In this case, we get the value f(q)=4 at two points q = -2 and q=+2.


X2 -4 to 4


2. Extreme Value Theorem

This theorem states that for a continuous function f(x) in the interval [a,b], there will be an absolute maximum and an absolute minimum. For a function f(x) that is continuous in the interval [a,b], there will be two points c and d such that f(c) will be absolute maximum and f(d) will be absolute minimum over [a,b].


Extreme Value Theorem


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