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Requiring that lim x a + f ( x ) = f ( a ) and lim x b f ( x ) = f ( b ) ensures that we can trace the graph of the function from the point ( a , f ( a ) ) to the point ( b , f ( b ) ) without lifting the pencil. If, for example, lim x a + f ( x ) f ( a ) , we would need to lift our pencil to jump from f ( a ) to the graph of the rest of the function over ( a , b ] .

Continuity on an interval

State the interval(s) over which the function f ( x ) = x 1 x 2 + 2 x is continuous.

Since f ( x ) = x 1 x 2 + 2 x is a rational function, it is continuous at every point in its domain. The domain of f ( x ) is the set ( , −2 ) ( −2 , 0 ) ( 0 , + ) . Thus, f ( x ) is continuous over each of the intervals ( , −2 ) , ( −2 , 0 ) , and ( 0 , + ) .

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Continuity over an interval

State the interval(s) over which the function f ( x ) = 4 x 2 is continuous.

From the limit laws, we know that lim x a 4 x 2 = 4 a 2 for all values of a in ( −2 , 2 ) . We also know that lim x −2 + 4 x 2 = 0 exists and lim x 2 4 x 2 = 0 exists. Therefore, f ( x ) is continuous over the interval [ −2 , 2 ] .

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State the interval(s) over which the function f ( x ) = x + 3 is continuous.

[ −3 , + )

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The [link] allows us to expand our ability to compute limits. In particular, this theorem ultimately allows us to demonstrate that trigonometric functions are continuous over their domains.

Composite function theorem

If f ( x ) is continuous at L and lim x a g ( x ) = L , then

lim x a f ( g ( x ) ) = f ( lim x a g ( x ) ) = f ( L ) .

Before we move on to [link] , recall that earlier, in the section on limit laws, we showed lim x 0 cos x = 1 = cos ( 0 ) . Consequently, we know that f ( x ) = cos x is continuous at 0. In [link] we see how to combine this result with the composite function theorem.

Limit of a composite cosine function

Evaluate lim x π / 2 cos ( x π 2 ) .

The given function is a composite of cos x and x π 2 . Since lim x π / 2 ( x π 2 ) = 0 and cos x is continuous at 0, we may apply the composite function theorem. Thus,

lim x π / 2 cos ( x π 2 ) = cos ( lim x π / 2 ( x π 2 ) ) = cos ( 0 ) = 1 .
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Evaluate lim x π sin ( x π ) .

0

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The proof of the next theorem uses the composite function theorem as well as the continuity of f ( x ) = sin x and g ( x ) = cos x at the point 0 to show that trigonometric functions are continuous over their entire domains.

Continuity of trigonometric functions

Trigonometric functions are continuous over their entire domains.

Proof

We begin by demonstrating that cos x is continuous at every real number. To do this, we must show that lim x a cos x = cos a for all values of a .

lim x a cos x = lim x a cos ( ( x a ) + a ) rewrite x = x a + a = lim x a ( cos ( x a ) cos a sin ( x a ) sin a ) apply the identity for the cosine of the sum of two angles = cos ( lim x a ( x a ) ) cos a sin ( lim x a ( x a ) ) sin a lim x a ( x a ) = 0 , and sin x and cos x are continuous at 0 = cos ( 0 ) cos a sin ( 0 ) sin a evaluate cos(0) and sin(0) and simplify = 1 · cos a 0 · sin a = cos a .

The proof that sin x is continuous at every real number is analogous. Because the remaining trigonometric functions may be expressed in terms of sin x and cos x , their continuity follows from the quotient limit law.

As you can see, the composite function theorem is invaluable in demonstrating the continuity of trigonometric functions. As we continue our study of calculus, we revisit this theorem many times.

The intermediate value theorem

Functions that are continuous over intervals of the form [ a , b ] , where a and b are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem    .

Questions & Answers

what is function.
Nawaz Reply
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
vector
what is the power rule
Vanessa Reply
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
Practice Key Terms 9

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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