1.4  Continuity

EXERCISE SOLUTIONS

For problems #1-3, let  g  be the piecewise defined function shown in the graph:

1.  Is g continuous at x=2?        Yes.

2.  Where does g have step discontinuities?       At  x=–1  and  x=1.

3.  Where does g have removable discontinuities?     At  x=–3 and  x=4.

4.  Classify any points of discontinuity of f over the graphed interval.

f  has a removable discontinuity at x=3.

Find any points of discontinuity of the functions in problems #5-10.   Specify whether or not the discontinuity is removable.

5.     

This function isn’t defined when x=1, but all other real numbers are in its domain.   This is an infinite discontinuity, and is not removable.   A graph of this function:

6.   ,   

so this function’s graph is the line y=x+5, but with a hole punched in the graph above x=3 (since the original function isn’t defined at this point).    The discontinuity is removable (by simply filling the hole).

7.  

The graph of this function is just the graph of  w = sin(t) / t,   but shifted to the right by p.   The function isn’t defined at x= p.   However, if we define its output to be 1 for input p, then the resulting function will equal its limit, and thus will be continuous at p.   So this is a removable discontinuity, as can be visually confirmed by looking at this function’s graph:

8.  

This function isn’t defined whenever its denominator is 0, which occurs whenever the cosine equals 1,  which occurs whenever q  is an integer multiple of 2p.   These are not removable discontinuities, as can be seen in the graph:

9.  

For this to be real-valued function, the radicand must be non-negative:

                                                                   

which will be true whenever s² is greater than or equal to 1.   The domain of this function is thus    The function is continuous at all points in its domain.

10.  

This function isn’t defined when t = 0, so the function is discontinuous there.   This point of discontinuity results from a “division by zero” singularity, and is not removable.   From the graph of the function one visually confirm that the discontinuity is not removable.

In creating this graph, I learned something new about graphing radical expressions in Maple…   My first attempts defined the function using fractional exponents, but the graphs just wouldn’t come out right.   Instead of using fractional exponents, the graphs come out much better if you use the surd function.  To define the surd function involves too far a detour into complex numbers.   Suffice it to say, instead of writing

                                                                         

use

surd(A,B)

absurd as it may seem!

So for example, to generate that last graph I used the command 

> plot(surd(t^2-1/t,5),t=-3..3,y=-   3..3,thickness=3,discont=true);

For problems 11-13, what value of a makes the function continuous at x=1?

These problems are all a simple matter of “making ends meet” !

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11.  

For the ends to match up at x=1, we set them equal and solve for a:

                                                                

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12.  

For the ends to match up at x=1, we set them equal and solve for a:

                                                                

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13.  

For the ends to match up at x=1, we set them equal and solve for a:

                                                             

14.   Use the Intermediate Value Theorem to show that the equation   cos x = x   must have at least one solution.

Consider the function  f(x) = x – cos x.   Then each solution to the given equation is a solution of f(x)=0.   This function f is continuous everywhere.    Since f(0) is negative and f(p/2) is positive x=1, the Intermediate Value Theorem implies that f has at least one root within the interval (0, p/2).