Continuity Worksheet

 

1Study the following functions and determine if they are continuous. If not, state where the discontinuities exist and what type they are:

1 Continuity Exercise

2Continuity Exercise

3Continuity Exercise

4Continuity Exercise

5Continuity Exercise

6Continuity Exercise

2Determine if the following function is continuous at x = 0.

Continuity Exercise

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

Continuity Exercise

4Are the following functions continuous at x = 0?

1 Continuity Exercise

2 Continuity Exercise

5Given the function:

Continuity Exercise

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.

6Determine if the following function is continuous. If not, state where the discontinuities exist or why the function is not continuous:

Continuity Exercise

7Determine if the following function is continuous: f(x) = x · sgn x.

8Determine if the following function is continuous at x = 0.

Continuity Exercise

9Determine the value of a to make the following function continuous.

Continuity Exercise

10The function defined by:

Continuity Exercise

is continuous on [0, ∞).

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


1

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

1 Continuity Problem

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.

2Continuity Problem

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.

Continuity Operations

Continuity Operations

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

3Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

The function is continuous.

4Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

The function has a jump discontinuity at x = 0 .

5Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

The function has a jump discontinuity at x = 1 .

6Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

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


2

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

Continuity Problem

f(0)=0

Continuity Operations

Continuity Operations

At x = 0, there is an essential discontinuity.


3

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

Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

At x = 1, there is a jump discontinuity.

Continuity Operations

Continuity Operations

Continuity Operations

At x = 2, there is a jump discontinuity.


4

Are the following functions continuous at x = 0?

1 Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

The function is continuous at x = 0.

2 Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

At x = 0, there is an essential discontinuity.


5

Given the function:

Continuity Problem

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

f(5) = 0.

Continuity Operations

Solve the indeterminate form.

Continuity Operations

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

Continuity Solution

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

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

Continuity Solution


6

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

Continuity Problem

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

Continuity Operations

Continuity Operations

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


7

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

Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Solution

The function is continuous.


8

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

Continuity Problem

The function Continuity Operations is bounded by Continuity Operations. therefore takes place:

límite, since any number multiplied by zero gives zero.

As f(0) = 0.

The function is continuous.


9

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

Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Solution


10

The function defined by:

Continuity Problem

is continuous on [0, ∞).

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

Continuity Operations

Continuity Operations

Continuity Solution




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