Exercise 1

Study the following functions and determine if they are continuous. If not, state where the discontinuities exist and what type they are:

1

2

3

4

5

6

Exercise 2

Determine if the following function is continuous at x = 0.

Exercise 3

Determine if the following function is continuous on (0,3). If not, state where the discontinuities exist and what type they are:

Exercise 4

Are the following functions continuous at x = 0?

1

2

Exercise 5

Given the function:

1 Prove that f(x) is not continuous at x = 5.

2Is there a continuous function which coincides with f(x) for all values with the exception x = 5? If so, determine the function.

Exercise 6

Determine if the following function is continuous. If not, state where the discontinuities exist or why the function is not continuous:

Exercise 7

Determine if the following function is continuous: f(x) = x · sgn x.

Exercise 8

Determine if the following function is continuous at x = 0.

Exercise 9

Determine the value of a to make the following function continuous.

Exercise 10

The function defined by:

is continuous on [0, ∞).

Determine the value of a that would make this statement true.

Solution of exercise 1

Study the following functions and determine if they are continuous. If not, state where the discontinuities exist:

1

The function is continuous at all points of its domain.

D = R − {−2,2}

The function has two points of discontinuity at x = −2 and x = 2.

2

The function is continuous at R with the exception of the values that annul the denominator. If this is equalled to zero and the equation is solved, the discontinuity points will be obtained.

x = −3; and by solving the quadratic equation: x=2−√3 and x=2+√3 are also obtained

The function has three points of discontinuity at x = −3, x = 2−√3 and x = 2+√3

3

The function is continuous.

4

The function has a jump discontinuity at x = 0 .

5

The function has a jump discontinuity at x = 1 .

6

The function has a jump discontinuity at x = 1/2 .

Solution of exercise 2

Determine if the following function is continuous at x = 0.

f(0)=0

At x = 0, there is an essential discontinuity.

Solution of exercise 3

Determine if the following function is continuous on (0,3). If not, state where the discontinuities exist and what type they are:

At x = 1, there is a jump discontinuity.

At x = 2, there is a jump discontinuity.

Solution of exercise 4

Are the following functions continuous at x = 0?

1

The function is continuous at x = 0.

2

At x = 0, there is an essential discontinuity.

Solution of exercise 5

Given the function:

1 Prove that f(x) is not continuous at x = 5.

f(5) = 0.

Solve the indeterminate form.

f (x) is not continuous at x = 5 because:

2Is there a continuous function which coincides with f(x) for all values with the exception x = 5? If so, determine the function.

If the function would be continuous, then the function is redefined:

Solution of exercise 6

Determine if the following function is continuous. If not, state where the discontinuities exist or why the function is not continuous:

The function f(x) is continuous for x ≠ 0. Therefore, study the continuity at x = 0.

The function is not continuous at x = 0, because it is defined at that point.

Solution of exercise 7

Determine if the following function is continuous: f(x) = x · sgn x.

The function is continuous.

Solution of exercise 8

Determine if the following function is continuous at x = 0:

The function is bounded by . therefore takes place:

, since any number multiplied by zero gives zero.

As f(0) = 0.

The function is continuous.

Solution of exercise 9

Determine the value of a to make the following function continuous:

Solution of exercise 10

The function defined by:

is continuous on [0, ∞).

Determine the value of a that would make this statement true.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.