A real-number function in one variable is a continuous function if and only if it is continuous at every point in its domain. Imagine a line or curve that’s smooth everywhere, with no gaps, no jumps, and no blow-ups in its domain. That’s what a continuous function looks like when graphed.
Linear functions
Alllinear functions are continuous. A real-number linear function L always has an equation of this form:
L (x)=ax+b
Lims(x) = 2
x 3 –
Lims(x) = 3
x 3 +
x 6
4
2
2 4 6
s(x)
Figure 2-6 Limits of a step function, s (x), at the point where x = 3.
Continuity of a Function 29
where x is the independent variable, a is a nonzero real number, and b can be any real number.
In rectangular coordinates, the graph of a linear function is a straight line that extends forever in two opposite directions. It never has a discontinuity.
Figure 2-7 shows four generic graphs of linear functions. The independent variable is on the horizontal axis, and the dependent variable is on the vertical axis.
Quadratic functions
All single-variable, real-number quadratic functions are continuous. The general form of a quadratic function Q is
Q (x)=ax2+bx+c
where x is the independent variable, a is a nonzero real, and b and c can be any reals. In rect-angular coordinates, the graph of a quadratic function is always a parabola that opens either straight up or straight down. The domain includes all reals, but the range is restricted to either the set of all reals greater than or equal to a certain absolute minimum, or the set of all reals less than or equal to a certain absolute maximum. There are never any gaps, jumps, or blow-ups in the graph within the domain.
Figure 2-8 shows four generic graphs of quadratic functions. The independent variable is on the horizontal axis, and the dependent variable is on the vertical axis.
Figure 2-7 Linear functions are always continuous. Imagine the lines extending smoothly forever from both ends.
Cubic functions
All single-variable, real-number cubic functions are continuous. The general form of a cubic functionC is
C (x)=ax3+bx2+cx+d
where x is the independent variable, a is a nonzero real, and b, c, and d can be any reals. In rect-angular coordinates, the graph of a cubic function looks like a badly distorted letter “S” tipped on its side, perhaps flipped over backward, and then extended forever upward and downward.
Unlike a quadratic function, which has a limited range with an absolute maximum or an absolute minimum, the range of a cubic function always spans the entire set of reals, although the graph can have a local maximum and a local minimum. The contour of the graph depends on the signs and values of a, b, c, and d. There are no gaps, blow-ups, or jumps within the domain.
Figure 2-9 shows four generic graphs of cubic functions. The independent variable is on the horizontal axis, and the dependent variable is on the vertical axis.
Polynomial functions
All single-variable, real-number polynomial functions are continuous. We can write the general form of an nth-degree polynomial function (let’s call it Pn) as follows, where n is an integer greater than 3, and x is never raised to a negative power:
Pn (x)=anxn+an−1xn−1+an−2xn−2+ · · · +a1x+b Figure 2-8 Quadratic functions are always continuous. Imagine
the curves extending smoothly forever from both ends.
Continuity of a Function 31
Here, x is the independent variable, a1,a2,a3, . . . , and an are called the coefficients, and b is called the stand-alone constant. The leading coefficient an, can be any real except 0. All the other coefficients, and the stand-alone constant, can be any real numbers.
The domain of an nth-degree polynomial function extends over the entire set of real numbers. If n is even, the range is restricted to either the set of all reals greater than or equal to a certain absolute minimum, or the set of all reals less than or equal to a certain absolute maximum. If n is odd, the range spans the entire set of reals, but there may be one or more local maxima and minima. The contour of the graph can be complicated, but there are never any gaps, blow-ups, or jumps within the domain.
Other continuous functions
Plenty of other functions are continuous. You can probably think of a few right away, remem-bering your algebra, trigonometry, and precalculus courses.
Discontinuous functions
A real-number function in one variable is called a discontinuous function if and only if it is not continuous at one or more points in its domain. Imagine a function whose graph is a line or curve with at least one gap, blow-up, or jump. That’s what a discontinuous function looks like when graphed.
Sometimes a discontinuous function can be made continuous by restricting the domain.
We eliminate all portions of the domain that contain discontinuities. For example, the function g∗, described earlier in this chapter and graphed in Fig. 2-5, is not continuous over the set of positive reals, because there’s a discontinuity at the point where x= 1. But we can make g∗
Figure 2-9 Cubic functions are always continuous. Imagine the curves extending smoothly forever from both ends.
continuous if we restrict its domain so that x > 1. We can also make it continuous if we restrict the domain so that 0 < x < 1, or so that x < 0. In fact, there are infinitely many ways we can restrict the domain and get a continuous function!
Here’s a challenge!
Look again at the function g (x)= 1/x that we worked with earlier in this chapter. If we define the domain ofg as the set of all positive reals except 0, is this function continuous? If we include x= 0 in the domain and give the function the value 0 there, calling the new function g#, is this function continuous?
Solution
As long as we don’t allow x to equal 0, the function g (x)= 1/x is both right-hand continuous and left-hand continuous at every point in the restricted domain. That means it’s a continuous function, even though its graph takes a huge jump. The blow-up in the graph does not represent a true discontinuity in g, because we don’t allow x= 0 in the domain.
Now suppose that we modify the function to allow x= 0 in the domain, calling the new function g# and including (0,0). Because neither the right-hand limit nor the left-hand limit is equal to 0 as x ap-proaches 0, the function g# is neither right-hand continuous nor left-hand continuous at the point where x= 0. Because of this single discontinuity, g# is not a continuous function.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not represent the only way a problem can be figured out. If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!
1. Find the limit of the infinite sequence
1/10, 1/102, 1/103, 1/104, 1/105, . . .
2. In a series, a partial sum is the sum of all the term up to, and including, a certain term.
As we include more and more terms in a series, the partial sum usually changes. Find the limit of the partial sum of the infinite series
1/10+ 1/102+ 1/103+ 1/104+ 1/105+ · · · as the number of terms in the partial sum approaches infinity.
3. Let x be a positive real number. Does the following limit exist? If so, find it. If not, explain why not.
Limx→∞ 1/x2
Practice Exercises 33
4. Let x be a positive real number. Does the following limit exist? If so, find it. If not, explain why not.
xLim→ +0 1/x2
5. Consider the base-10 logarithm function (symbolized log10). Sketch a graph of the functionf (x)= log10x for values of x from 0.1 to 10, and for values of f from −1 to 1.
Determine
xLim→ −3 log10x 6. Look again at f (x)= log10x. Find
xLim→ +3 log10x
7. Based on the answers to Probs. 5 and 6, can we say that f (x)= log10x is continuous at the point where x= 3? If so, why? If not, why not?
8. Is the function f (x)= log10x continuous over the set of positive reals? If not, where are the discontinuities? Is this function continuous over the set of nonnegative reals (that is, all the positive reals along with 0)? If not, where are the discontinuities?
9. Sketch a graph of the absolute-value function (symbolized by a vertical line on either side of the independent variable) for values of the domain from approximately −6 to 6.
Is this function continuous over the set of all reals? If not, where are the discontinuities?
10. Sketch a graph of the trigonometric cosecant function for values of the domain between, and including, −3p radians and 3p radians. Is this function continuous if we restrict the domain to this closed interval? If not, where are the discontinuities?
Remember that the cosecant (symbolized csc) of a quantity is equal to the reciprocal of the sine (symbolized sin). That is, for any x,
cscx= 1 / (sin x)
All single-variable linear functions have straight-line graphs. The slope of such a graph can be found easily using ordinary algebra. In this chapter, we’ll learn a technique that allows us to find the slope of a graph whether its function is linear or not.