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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

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.

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:

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

Let's see an example:

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

We will now look at two important theorems pertaining to continuous functions: the intermediate value theorem and the extreme 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*.

Taking the example of the previous function,

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

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.

Now, when it comes to the **extreme value theorem**, which 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*], as seen here:

Taking the same example as before, in the range [1,2], the absolute maximum and minimum for the function will be 4 and 1 at points 2 and 1, respectively which can be seen in this graph:

In this lesson, you learned about the **continuous functions** and the conditions that a function has to fulfill in order to be one. It's basically a function that has an unbroken, continuous graph between two points. Then we looked at two important theorems pertaining to continuous functions:

**Intermediate value theorem**: states that for a function*f(x)*that is continuous between [*a,b*], and there is a number*p*between*f(a)*and*f(b)*, then there must be at least one value*q*between*a*and*b*, such that*f(q) = p*

**Extreme value theorem**: states that for a continuous function*f(x)*in the interval [*a,b*], there will be an absolute maximum and an absolute minimum

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

- Squeeze Theorem: Definition and Examples 5:49
- Using a Graph to Define Limits 5:24
- Understanding Limits: Using Notation 3:43
- One-Sided Limits and Continuity 4:33
- How to Determine the Limits of Functions 5:15
- Understanding the Properties of Limits 4:29
- Continuity in a Function 5:37
- Discontinuities in Functions and Graphs 6:26
- Regions of Continuity in a Function 5:22
- Continuous Functions Theorems 3:17
- Intermediate Value Theorem: Examples and Applications 6:30
- Go to Limits of Sequences & Functions

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