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Determine where the function f ( x ) = { π x 4 ,    x < 2 π x ,      2 x 6 2 π x ,    x > 6 is discontinuous.

x = 6

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Determining whether a function is continuous

To determine whether a piecewise function    is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.

Given a piecewise function, determine whether it is continuous.

  1. Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?
  2. For each boundary point x = a of the piecewise function, determine if each of the three conditions hold.

Determining whether a piecewise function is continuous

Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.

f x = { sin ( x ) , x < 0 x 3 , x > 0

The two functions composing this piecewise function are f ( x ) = sin ( x ) on x < 0 and f ( x ) = x 3 on x > 0. The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,

At x = 0 , let us check the three conditions of continuity.

Condition 1:

f ( 0 )  does not exist . Condition 1 fails .

Because all three conditions are not satisfied at x = 0 , the function f ( x ) is discontinuous at x = 0.

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Access these online resources for additional instruction and practice with continuity.

Key concepts

  • A continuous function can be represented by a graph without holes or breaks.
  • A function whose graph has holes is a discontinuous function.
  • A function is continuous at a particular number if three conditions are met:
    • Condition 1: f ( a ) exists.
    • Condition 2: lim x a f ( x ) exists at x = a .
    • Condition 3: lim x a f ( x ) = f ( a ) .
  • A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
  • A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See [link] .
  • Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See [link] and [link] .
  • For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See [link] and [link] .

Section exercises


State in your own words what it means for a function f to be continuous at x = c .

Informally, if a function is continuous at x = c , then there is no break in the graph of the function at f ( c ) , and f ( c ) is defined.

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State in your own words what it means for a function to be continuous on the interval ( a , b ) .

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For the following exercises, determine why the function f is discontinuous at a given point a on the graph. State which condition fails.

f ( x ) = ln   |   x + 3   | , a = 3

discontinuous at a = 3 ; f ( 3 ) does not exist

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f ( x ) = ln   |   5 x 2   | , a = 2 5

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f ( x ) = x 2 16 x + 4 , a = 4

removable discontinuity at a = 4 ; f ( 4 ) is not defined

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f ( x ) = x 2 16 x x , a = 0

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f ( x ) = { x ,     x 3 2 x , x = 3   a = 3

Discontinuous at a = 3 ; lim x 3 f ( x ) = 3 , but f ( 3 ) = 6 , which is not equal to the limit.

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Questions & Answers

How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
lol...it gets better
I've been struggling so much through all of this. my final is in four weeks 😭
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
more than 6000
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
can I see the picture
Zairen Reply
Practice Key Terms 4

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