Continuity: the Concept and Application of Continuity in Mathematics (Part III)

Continuity (Part III)

Composition of functions is the process of combining two or more functions to form a new function. The composition of two functions can be represented by a composite function F(x) = g(f(x)), such that the composite function takes inputs and returns outputs.

Intermediate Value Theorem for Continuous Functions: If a function is continuous on an interval, then it has all possible values between two given values. The theorem is useful in proving that certain functions are continuous.

You can prove this theorem using the fact that f(x) = g(x) if and only if there exists a number c such that g(c) = f(c).

Using this composition of functions, we can generate complicated functions from simpler ones. Here we have the following theorem about continuous functions.

If a function f is continuous at a point c, and another function g is continuous at the value of f(c), then the composition function g circle f will be continuous at c.

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Let's go back to the graph with this picture we saw earlier. If machine f is continuous at x and machine g is continuous at here, then the combined machine will be continuous at this point. The proof of this theorem is as follows.

According to the property of continuous functions, we can compute the limit of this composition function when x tends to c. This is equal to x tends to c times g of f(x). It is a definition of composition.

As g is continuous at f(c), we can take the limit inside the equals sign.

This limit is equal to g times the limit x tends to c of f(x). Now, since f is continuous at c, this means that this expression is just equal to f(c). And that's just equal to the combined machine at the value c.

Since the limit of this function as x approaches c exists, and is equal to the value of the function at the point c according to the definition that this composition function is continuous at c.

So we have the following formula. The last thing we will discuss is a very useful theorem about continuous functions, which we call the Intermediate Value Theorem for Continuous Functions.

If function f is continuous on the closed interval [a,b], then for any value I between f(a) and f(b), there exists some c between a and b such that f(c) equals to I.

Suppose that this is a and that this is b, so the theorem states that any horizontal line intersecting these two points will cut the graph at least one more point.

The Intermediate Value Theorem says that if a function is continuous at some point c, then the function will have an inverse value at c. This means that every horizontal line between f(a) and f(b) will intersect the graph of the function at some point c, where it has an inverse value. Therefore, no gaps exist in the graph of a continuous function.

As an application, we can show that there is a root of this polynomial between zero and two.

If you draw the graph of this function, the point zero, f(0) equals to minus five, and f(2) equals 51. You can compute this very easily. This one is below zero; this point is above

zero. So if you draw a horizontal line as zero, this horizontal line must cut this function at some point c, which means that f(c) = 0.

Now, since c is between zero and two, we can conclude that c satisfies the equation f(c)=0. Therefore, c is a root of this polynomial. Now, let's look at an example of a continuous function that does not extend continuously at a point.

The function sine x over x, shown here, has a graph that looks like this.

The function is undefined at the point x equals zero, because if you plug in x equals zero in the top and bottom, you get the zero over zero which is meaningless.

Thus, we see that the graph of the function has a hole at x = 0. This undesirable situation can be remedied by filling in the hole with a value of one.

The function F is defined as follows: When x is not equal to zero, the original function here; when x equals zero, we define F(x) = 1. Now after filling up the hole, we get a continuous function.

This is how we attempt to remedy the discontinuity of a function.

To fill up the hole, we first make it into a continuous function. Here's how we do that: The function must have a limit at the point c and then we define the value of the function at this point to be just the limit of the function.

And after that we have this continuous extension to a point.