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CONTINUITY OF FUNCTIONS OF ONE VARIABLE

The following problems involve the CONTINUITY OF A FUNCTION OF ONE

VARIABLE. Function y = f(x) is continuous at point x=a if the following threeconditions are satisfied :

i.) f(a) is defined ,

ii.) exists (i.e., is finite) ,

and

iii.) .

Function fis said to be continuous on an interval Iiffis continuous at each

point x in I. Here is a list of some well-known facts related to continuity :

1. The SUM of continuous functions is continuous.

2. The DIFFERENCE of continuous functions is continuous.

3. The PRODUCT of continuous functions is continuous.

4. The QUOTIENT of continuous functions is continuous at all

points x where the DENOMINATOR IS NOT ZERO.

5. The FUNCTIONAL COMPOSITION of continuous functions is

continuous at all points x where the composition is properly defined.

6. Any polynomial is continuous for all values of x.

7. Function ex and trigonometry functions and are continuous for

all values ofx.

Most problems that follow are average. A few are somewhat challenging. All

limits are determined WITHOUT the use of L'Hopital's Rule. If you are going totry these problems before looking at the solutions, you can avoid common mistakes

by using the above step-by-step definition of continuity at a point and the well -

known facts, and by giving careful consideration to the indeterminate form

during the computation of limits. Knowledge of one-sided limits will be required.

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For a review of limits and indeterminate forms click here.

o PROBLEM 1 : Determine if the following function is continuousat x=1 .

o PROBLEM 2 : Determine if the following function is continuousat x=-2 .

o PROBLEM 3 : Determine if the following function is continuousat x=0 .

o PROBLEM 4 : Determine if the function is continuousat x=-1 .

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o PROBLEM 5 : Check the following function for continuity at x=3and x=-3 .

o PROBLEM 6 : For what values ofx is the

function continuous ?

o PROBLEM 7 : For what values ofx is thefunction continuous ?

o PROBLEM 8 : For what values ofx is the functioncontinuous ?

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o PROBLEM 9 : For what values ofx is the functioncontinuous ?

o PROBLEM 10 : For what values ofx is the

function continuous ?

o PROBLEM 11 : For what values ofx is the following functioncontinuous ?

o PROBLEM 12 : Determine all values of the constant A so that thefollowing function is continuous for all values of x .

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o PROBLEM 13 : Determine all values of the constants A and B so thatthe following function is continuous for all values of x .

o PROBLEM 14 : Show that the following function is continuous for allvalues ofx .

o PROBLEM 15 : Let

Show that fis continuous for all values ofx . Show that fis

differentiable for all values ofx, but that the derivative, f' , is NOT

CONTINUOUS at x=0 .