Continuity and discontinuity of functions illustrated by their graphs

1. Consider the function $x^2$, and let

   $$y = x^2$$ (3.4)

   
   
   If we assume values for x and calculate the corresponding values of y, we can plot a series of points. Drawing a smooth line free-hand through these points: a good representation of the general behavior of the function may be obtained. This picture or image of the function is called its graph. It is evidently the locus of all points satisfying equation (3.4).

   Figure 3.1: The parabola $y=x^2$.

   

   It is very easy to create the above plot in Sage, as the example below shows:

   [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
   
   sage: P = plot(x^2,-2,2)
   sage: show(P)
   

   Such a series or assemblage of points is also called a curve. Evidently we may assume values of $x$ so near together as to bring the values of $y$ (and therefore the points of the curve) as near together as we please. In other words, there are no breaks in the curve, and the function $x^2$ is continuous for all values of $x$.

2. The graph of the continuous function $\sin x$, plotted by drawing the locus of $y = \sin\, x$,

   Figure 3.2: The sine function.

   

   It is seen that no break in the curve occurs anywhere.

3. The continuous function $exp(x) = e^x$ is of very frequent occurrence in the Calculus. If we plot its graph from
   $$y= e^x, \qquad (e = 2.718\cdots),$$
   we get a smooth curve as shown.

   Figure 3.3: The exponential function.

   

   From this it is clearly seen that,

   (a)

   when $x = 0$, $\lim_{x \to 0} y (= \lim_{x \to 0} e^x) = 1$;

   (b)

   when $x > 0$, $y (= e^x)$ is positive and increases as we pass towards the right from the origin;

   (c)

   when $x < 0$, $y (= e^x)$ is still positive and decreases as we pass towards the left from the origin.

4. The function $\ln \, x = \log_e\ x$ is closely related to the last one discussed. In fact, if we plot its graph from
   $$y = \log_e\ x,$$
   it will be seen that its graph has the same relation to $OX$ and $OY$ as the graph of $e^x$ has to $OY$ and $OX$.

   Figure 3.4: The natural logarithm.

   

   Here we see the following facts pictured:

   (a)

   For $x=1$, $\log_e\ x = \log_e\ 1 = 0$.

   (b)

   For $x > 1$, $\log_e\ x$ is positive and increases as $x$ increases.

   (c)

   For $1 > x > 0$, $\log_e\ x$ is negative and increases in numerical value as $x$, that is, $\lim_{x \to 0} \log\ x = -\infty$.

   (d)

   For $x \le 0$, $\log_e\ x$ is not defined; hence the entire graph lies to the right of $OY$.

5. Consider the function $\frac{1}{x}$, and set
   $$y = \frac{1}{x}$$
   If the graph of this function be plotted, it will be seen that as $x$ approaches the value zero from the left (negatively), the points of the curve ultimately drop down an infinitely great distance, and as $x$ approaches the value zero from the right, the curve extends upward infinitely far.

   Figure 3.5: The function $y=1/x$.

   

   The curve then does not form a continuous branch from one side to the other of the axis of $y$, showing graphically that the function is discontinuous for $x = 0$, but continuous for all other values of $x$.

6. From the graph (see Figure 3.6) of
   $$y = \frac{2x}{1 - x^2}$$
   it is seen that the function $\frac{2x}{1 - x^2}$ is discontinuous for the two values $x = \pm 1$, but continuous for all other values of $x$.

   Figure 3.6: The function $y=2x/(1-x^2)$.

   

7. The graph of
   $$y = \tan\ x$$
   shows that the function $\tan x$ is discontinuous for infinitely many values of the independent variable $x$, namely, $x = \frac{n\pi}{2}$, where $n$ denotes any odd positive or negative integer.

   Figure 3.7: The tangent function.

   

8. The function $\arctan\ x$ has infinitely many values for a given value of $x$, the graph of equation
   $$y = \arctan\ x$$
   consisting of infinitely many branches.

   Figure 3.8: The arctangent (or inverse tangent) function.

   

   If, however, we confine ourselves to any single branch, the function is continuous. For instance, if we say that $y$ shall be the arc of smallest numerical value whose tangent is $x$, that is, $y$ shall take on only values between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, then we are limited to the branch passing through the origin, and the condition for continuity is satisfied.

9. Similarly, $\arctan \frac{1}{x}$, is found to be a many-valued function. Confining ourselves to one branch of the graph of
   $$y = \arctan\ \frac{1}{x},$$
   we see that as $x$ approaches zero from the left, $y$ approaches the limit $-\frac{\pi}{2}$, and as $x$ approaches zero from the right, $y$ approaches the limit $+\frac{\pi}{2}$. Hence the function is discontinuous when $x = 0$. Its value for $x = 0$ can be assigned at pleasure.

   Figure: The function $y=\arctan(1/x)$.

   

10. A piecewise defined function is one which is defined by different rules on different non-overlapping intervals. For example,
    $$f(x) = \left\{ \begin{array}{ll} -1, & x<-\pi/2,\\ \sin(x), &\pi/2\leq x\leq \pi/2,\\ 1, &\pi/2<x. \end{array}\right.$$
    is a continuous piecewise defined function.

    Figure 3.10: A piecewise defined function.

    

    For example,

    $$f(x) = \left\{ \begin{array}{ll} -1, & x<-2,\\ 3, &-2\leq x\leq 3,\\ 2, &3<x. \end{array}\right.$$
    is a discontinuous piecewise defined function, with jump discontinuities at $x=-2$ and $x=3$.

    Figure 3.11: Another piecewise defined function.

    

Functions exist which are discontinuous for every value of the independent variable within a certain range. In the ordinary applications of the Calculus, however, we deal with functions which are discontinuous (if at all) only for certain isolated values of the independent variable; such functions are therefore in general continuous, and are the only ones considered in this book.