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Continuous Functions Theorems

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  • 0:03 Continuous Functions
  • 0:57 Theorems of Continuous…
  • 2:33 Lesson Summary
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Instructor: Usha Bhakuni

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

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

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: the intermediate value theorem and the extreme value theorem.

1. Intermediate Value Theorem

For the 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 like this:


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, as we can see in this graph.


X2 -4 to 4


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