Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)

Chapter 1: Differentiation; Section 2: Real Functions; page 1

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Basics

Definition of a function

A real function defines a rule of correspondence between a domain of real numbers, and another domain of real numbers. Since we are only interested in real numbers, for the time being, for abreviation we are often going to omit the word "real". Let's call one of the domains the domain of independent variables (usually but not necessarely denoted by  x) and the other - the domain of the dependent variables (usually but not necessarely denoted by  y). The rule of correspodence is generally denoted by  f  (for function) which we express in the following way:

One can consider the function as an operation transforming an argument, which in this case is x. In the more general case the argument of a function can also be a mathematical expression of the independent variable. For a function denoted by name e.g.  fun , we'll use the notation  fun x  , where the name of the function is expressed with regular letters and for the variable  x - italic fonts are used. Only if the argument is a mathematical expression will we use parentheses. E.g. if the expression is  x+1 , then  fun (x+1)  will be used.

According to the above definition there is no restriction on the domains of the independent and dependent variables. As an example we can use the sequence of natural numbers as the independent variables, and the rational numbers as the dependent - and use a rule of relating between them ( reminder: Fig. Countability ).

Another example: for the function  f(x) = x²

the domain of the independent variable includes all the real numbers, while the dependent variable is limited to the domain of the non negative real numbers.

The third example will be:

which is limited to the non-negative domain of the independent variables. On the other hand the dependent variable is doubly-valued. In order to keep the dependent variable single valued, we consider the function as consisting of two separate branches one with non-negative dependent values, and one with non-positive dependent values, each branch consisting of a single valued function. As we'll see there are also cases with multiple-valued (more then two) dependent variables, which can be separated into different single-valued branches. Usually (but not necessarily) the study of one of them will suffice, and could be generalized to the rest.

The domain of a variable in empirical sciences can be restricted sometimes by its properties. For instance the absolute temperature, the pressure etc. are defined only for non negative values.

Graphical presentation

Any function can be presented in graphical form by use of two axes of real numbers called abscissa (horizontal) and ordinate (vertical). The abscissa belongs to the independent variable (e.g. x) and is drawn with numbers increasing from left to right. The ordinate belongs to the dependent variable (e.g. y) and is drawn with numbers increasing from bottom up. This represents the Cartesian system of coordinates.

The point corresponding to zero on both axes is called the origin of the coordinates, and it is customary (but not necessary) to draw the axes intersecting at the origin.

The scales used for the abscissa and for the ordinate do not have to be equal. They are usually chosen in a way to display the graphics of the function in a more convenient way.

As an example the graphical presentation of the function

is given and explained on Fig. Cartesian coordinates.

Limits and Continuity

An interval between a and b (a<b) is the set of all (real) numbers between these values. An interval that includes the end points (a and b) is called closed and is denoted by the square brackets [a,b]. If the interval does not include the end points in, is called open, and is denoted by round brackets (a,b). An interval can also be open on left and closed on right (a,b] or opposite [a,b).

If a function  f(x)  is single-valued for all values of  x  in an interval except or not for the value  $x=x_0$ , we say that

if for any sequence of x values within the interval converging to, but not including  $x_0$ , the corresponding sequence of the  f(x)  values converges to L .

In a similar way by using the limit of a sequence to infinity, one defines also

which of course could have also negative sign, depending on the circumstances. Irrespectively of the sign, the locations corresponding to the function diverging to infinity will be marked graphically by vertical dashed straight lines intersecting the x axis at the corresponding values.

If the independent variable x is not restricted to the right, meaning that one can make a sequence of  x  values diverging to infinity, one can define the limit

and calculate it, if it exists. This kind of limit can be expressed simply by a sequence with x taking the values of consecutive natural numbers. This can give very usefull information about the behaviour of the function. By the same token, if appropriate, one can also use the limit of  $x\to -\infty$ .

In some cases one imposes the following conditions on the sequence for the limit at point  $x=x_0$ : limit from the right (denoted by  $x\to x_{0+}$), if the sequence is restricted to values$>x_0$  and limit from the left (denoted by $x\to x_{0-}$), if the sequence is restricted to values$<x_0$ . Of course, if  $x_0$  is the end point of the interval, the only possible limit is from one side.

In the case that

A function is continuous at a given point  $x_0$ , if the function is defined there (but is not infinite) and if

A function is continuous in an interval, if it is continuous at each point of this interval. To put it bluntly: A function is continuous if you can draw it without taking your pen off the paper.

As an example, if a function is defined only for irrational numbers of the independent variable, it cannot be continuous, since any sequence for the calculation of a limit is resticted to irrational values. If we use the blunt definition, in order to draw the function you have to raise your pen everytime you pass through a point corresponding to a rational number of the independent variable.

From this point on, we are going to deal mostly with continuous functions in the domain in which they are defined, which may or may not include some points of discontinuity. The domain could include all the real numbers, or could consist of one or more separate intervals.

Examples

We are going to analyse a couple of function on basis of the material learned at this page.

Example 1

The function is continuous in two separate intervals  x < 0  and  x > 0 .

The limit of the function at point  x = 0  from the left is  −1  and from the right is  +1 . There is not a possible value of y in order to make the function continuous at that point.

Example 2

For  $-\infty$  the limit exists and is $\underset{x\to -\infty }{\lim }f\left(x\right)=-1$ . This can be obtained e.g. by substituting x by −n , where n is a natural number: $$\underset{x\to -\infty }{\lim }\frac{1-x}{1+x}=\underset{n\to \infty }{\lim }\frac{1+n}{1-n}=\underset{n\to \infty }{\lim }\frac{n\left(\frac{1}{n}+1\right)}{n\left(\frac{1}{n}-1\right)}=-1$$

The function is continuous for  x < −1 .

For  x = −1  the function diverges to infinity and therefore will be marked in its graphical display by a dashed vertical line. From the left:

$$\underset{x\to -1_{-}}{\lim }\frac{1-x}{1+x}=-\infty$$ and from the right: $$\underset{x\to -1_{+}}{\lim }\frac{1-x}{1+x}=+\infty$$
therefore at the point  x = −1 , the function is not continuous .

In the interval  −1 < x < 0  the function is continuous.

At  x = 0  the limits of the function from the left and from the right are equal to one (=1), and therefore the function is continuous there with value  $f\left(0\right)=1$ .

For  $x\ge 0$  the function is constant and $f\left(x\right)=1$ .

An approximate graphic display, where points of the function are connected by straight lines is presented in Fig. Continuity (not animated) . Notice that the abscissa and ordinate are not drawn with the same scale.

Exercises

Study the following functions according to the guidelines:

• Give the domain of the independent variable in which the function is continuous.
• Look closely at each point of discontinuity. Could one define a limit? If yes, what is the limit? Could the function be continuous at this point?
• If appropriate, what is the behaviour of the function at infinity?
• Give an approximate graphical presentation of the function in the most interesting region (at and near the finite extremities of the continuous intervals of x). Connect points with straight lines. Mark with a dashed vertical line the divergence of the function to infinity.

Exercise 1.

Exercise 2.

Exercise 3.

Exercise 4.

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