Continuity.mws

Limits and Derivatives: Continuity

Vocabulary

continuous, discontinuous, continuous on an interval, removable discontinuity, infinite discontinuity, jump discontinuity, Intermediate Value Theorem

Objectives

Lecture Outline

In everyday language, something is said to be continuous if it has uninterrupted extension in space or time. That is, something has the property of continuity if there are no abrupt changes in it. The mathematical notion of continuity is similar. Let us begin with the mathematical definition of continuity.

Definition of Continuity

A function f is continuous at a number a if Limit(f(x),x = a) = f(a) .

The definition given above actually contains three assertions within it, neatly packaged into one limit statement. Three requirements must be met if a function f is to be continous at a :

If one of the three conditions fails at a point a , then we say that f is discontinuous at a .

Types of Discontinuities

In this course we will encounter three basic ways in which a function may fail to be continuous.

Removable Discontinuity

Consider the function f(x) = (x^2-4)/(x-2) . One may immediately observe first that a = 2 is not in the domain of f and second that after factoring the numerator and cancelling common factors, the function may be re-expressed as f(x) = x+2 for x not equal to 2. The graph of f(x) looks like the graph of the line y = x+2 with the point (2,4) removed.

We say that f has a removable discontinuity at a = 2 since it is possible to fill in the missing point (2,4) on the graph of f so as to obtain a new function which is continuous at a = 2 . By filling in the missing point (the gap in the graph), we remove the discontinuity.

Infinite Discontinuity

Let us now consider the function g(x) = (x-3)^2/((x+1)^2) . Notice that a = -1 is the only real number not in the domain of f . Hence g is continuous everywhere except at a = -1 . Consider the graph of the function:

> plot((x-3)^2/(x+1)^2,x=-3..1,y=-5..50,discont=true);

[Maple Plot]

We see that Limit(g(x),x = -1) = infinity . In this case, g is discontinuous at a = -1 by virtue of the fact that g increases without bound as x approaches -1. We say that g has an infinity discontinuity at a = -1 .

Jump Discontinuity

Often when considering piecewise defined functions, there may be "jumps" in their graphs. Consider the graph of the function h(x) = abs(x)/x , for example.

> plot(abs(x)/x,x=-2..2,discont=true);

[Maple Plot]

Here it is noticed that as x moves across 0 from left to right, h(x) jumps from the value -1 to 1. That is, h abruptly changes as x moves from negative values to positive values. Further, h(0) is undefined and Limit(h(x),x = 0) does not exist, so h fails to be continuous at 0. We say that h has a jump discontinuity at 0.

Continuity on an Interval

A function is continuous on an interval I if it is continuous at every point in I. Notice that we must be careful here when dealing with intervals containing endpoints where the function is not defined outside the interval. In these cases, we must introduce the ideas of continuity from the right and continuity from the left .

Definition of one-sided continuity

A function f is continuous from the right at the point a if Limit(f(x),x = a,right) = f(a) and f is continuous from the left at the point a if Limit(f(x),x = a,left) = f(a) .

If a function is defined on only one side of an endpoint of an interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left depending on the given situation.

Common Continuous Functions

Most of the elementary functions which have been introduced so far are continuous wherever they are defined. Hence the following types of functions are continuous at every number in their domains: root functions, polynomials, rational functions, trigonometric functions, and exponential functions. Moreover, the inverse function of any continuous function is continuous.

Composition of Continuous Functions

One can build complicated continuous functions by composing elementary continuous functions with one another by the following theorem.

Theorem (The Composition of Continuous Functions is Continuous)

If g is continuous at a and f is continuous at g(a) , then the composite function given by f(g(x)) is continuous at a .

One of the basic properties of continuous functions is the subject of the following theorem.

The Intermediate Value Theorem (IVT)

If f is continuous on the closed interval [ a , b ] and N is any number between f(a) and f(b) , then there exists a number c between a and b such that f(c) = N .

Example: We show that x^3+2*x+2 = 0 has a real root between -1 and 0. Recall that a root of an equation is a value of the variable which satisfies the equation. Let f(x) = x^3+2*x+2 . To show that our given equation has a real root between -1 and 0, we must show that there is a real number c such that f(c) = 0 . We do this by appropriately applying the Intermediate Value Theorem.

We note that f(x) , being a polynomial, is continuous everywhere. In particular, it is continuous on the closed interval [-1,0]. Evaluating f at the endpoints of this interval, we see that f(-1) = -1 and f(0) = 2 . Hence the number N = 0 is between f(-1) and f(0) . The conclusion of the Intermediate Value Theorem is that there must be a number c between -1 and 0 such that f(c) = 0 . We have shown that there is a real root between -1 and 0.

Maple allows us to verify what we have shown using the IVT. We plot the graph of f(x) = x^3+2*x+2 and verify that the graph does indeed cross the x-axis at some number between -1 and 0.

> f:=x->x^3+2*x+2;

f := proc (x) options operator, arrow; x^3+2*x+2 en...

> plot(f(x),x=-1..0);

[Maple Plot]

The Intermediate Value Theorem is a powerful theoretical tool in the calculus and one of the first nontrivial results we have seen so far. As an existence theorem, it does not give us specific information about the number c , but as we will see, knowledge of existence is enough for some interesting results later on.