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

Continuity(Part II)

A polynomial is an expression that has a degree of more than 1. It can be written as a sum of products of powers of variables and constants. A polynomial can also be defined as an algebraic expression which involves only terms with whole number exponents and whose highest exponent is finite.

Now we come to some properties of continuous functions. As we know now, continuity depends on the concept of limits. For determining these limits, we have a set of limit laws that can help us determine the limits of combinations of functions.

Continuity functions have the following properties: for example, if f and g are both continuous at a certain point x equals c, now at this point x equals c we see that f plus g is also continuous at that point and f minus g is also continuous at that point.

1. f+g
2. f-g

This result follows from the limit law applying to f plus g and f minus g.

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The same way, if we multiply the two functions f and g, this function will also be continuous at c.

3. fg

If we multiply a constant a to the function f, then the resulting function af will also be continuous at the point c.

4. af for any constant a

The continuity of a function f at a point c follows from the same rule for the limits. Similarly, if we divide f by g, then this will also be continuous at c.

Note that there is an exception to this rule.

If g(c) equals zero, then f over g is not defined. In other words, the rule cannot be applied if g(c) is equal to zero.

Similarly, we can apply the hower root to the function f and see that f to the power m over n would also be continuous.

The function f(x) equals to x is continuous at every point x. Really simple.

It happens because if you draw the graph of a function where f(x) equals square root of x, you will see that it is a very simple straight line passing through the origin making 45 degrees with the positive x axis. At x equals to zero, the function f(x) equals 0 but is only right-continuous because the function is not defined when x is negative.

Because you have only positive values for x, you can only have a right-continuous function at the point x equals to zero. There are many other functions which are continuous.

Instead of just a straight line, we have two x plus three and a straight line that is continuous. Seven x squared minus x is also continuous. In general, we can look at functions with degree n, degree n minus one, and so on.

A polynomial is a function of the form ƒ(x) = a0+a1x+a2x2+...an, where each ai is a constant. In general, all polynomials are continuous functions. On top of that, there are more functions which are also continuous.

For example, the trigonometric functions you've seen in high school--the sine function, cosine functions --are continuous. But what about the tangent function? The tangent function is defined as the sine function divided by the cosine function.

Now, as you do division, remember that most of the time it is continuous except at those places where the denominator equals zero.

Therefore, for the tangent function, cosine is equal to zero only when x equals 2p or 2p plus multiples of pi. At these places, the function is not continuous.

However, at other places the tangent function is continuous. Furthermore, we will see later on that the exponential function and also the logarithmic functions are always continuous in their domains of definition.

We can also compose or make more complicated functions from such functions. For example, the sine of the square root of x squared plus one is continuous, as is exponential of the square root of x.

We can see that the function is continuous for x greater than the root of zero because the square root of x is continuous for x > 0.

We also know that the polynomial x squared plus one, written as (x)² + 1, is continuous. How can we see that the square root of this polynomial is also continuous? Well, we can understand this as a composition of two functions.

I will attempt to explain this concept of the composition of two functions in the following way. Suppose the function f(x) equals x squared plus 1, which is just like a machine that produces output when we put an input into it.

So. This machine takes in x as input, and generates x squared plus one as output.

Now suppose that I have an input of three into the machine, then the output is 10. Equals three squared plus one.

Suppose I have another machine in here labeled g(x).

What it does is the following.

Whatever you input into the machine, it will generate as output a square root of that expression.

When I enter the expression x squared plus one, the output is the square root of the input. For example, if I enter number 10, the output is the square root of 10.

If we combine these two machines in this particular order, we can consider them as one single machine.

One can regard the machine described above as a single unit, because it performs only one task: taking an input and outputting a result.

Therefore, we can regard it as a single machine, covering up the intermediate part. The machine takes x as input, and outputs the square root of x squared plus one.

You input three, and the output is the square root of 10. The formula for this machine is square root of x squared + 1.

The combined machine g circle f is the result of performing machine f and then performing machine g. It is written in this symbol: g circle f.