Continous function

Continous function
Written by: Jack Methew

Jack Methew knows that successful students become successful adults. This is her 15th year at Edison Elementary School and her 10th year teaching fourth grade. So far, fourth grade is her favorite grade to teach! Mrs. Carroll was the 2011 Newell Unified School District Teacher of the Year, and received her National Board Certification in 2013. She loves science and majored in biology at Arizona State University, where she also earned her teaching credential and Master of Education degree. Mrs. Carroll is excited to begin the best year ever!

See Article History

Continuity of Function

A function is said to be continuous if it has no breakable graph

Graphical Representation:

Continuity Function

Fact: Here you can see the graph has no breaks or holes in it which means it is a continuous function and it also shows there is no missing point between values.

Question: What does discontinuity mean then?

Answer: It shows holes or breaks in the graph.

Discontinuous Function

A type of function which has not continuous graph which indicates hole or break in graph

Few Quick Review:

  • As,there is hole and jump so these are discontinuous function

    Continuity Function

  • Asymptote also show discontinuity of function

    Continuity Function

Domain of Function

Domain and range are two important portion in functions

Continuity Function

Example:

$$\frac{1}{x3-1}$$

$$\text{When}\;x=1$$

Continuity Function

$$\frac{1}{x3-1}=\frac{1}{0}=\text{undefined}$$

So,we can say that this function is discontinuous at x=1

So

$$F(x)=\frac{1}{x3-1}$$

is not continuous(discontinuous) at every real number.

Let us change domain to x>1

Continuity Function

So ,here g(x) is continuous also it does not contain the value of x=1 so it is continuous

Fact: If a function is continuous in its domain is called continuous function.

Formal Definition of continuity of function

A function is said to be continuous at x=a if

  • If f(x) is defined
  • $$\lim\limits_{x \to a}⁡f(x)=f(a)$$
  • The limit of f(x) is f(c) as x approaches to a

Limit from both sides:

Continuity Function

Continuity Function

  • If limit of left hand side

    ≠ to limit of right hand side so limit does not exist

  • And this is true for every value of a in its domain

Make Sure two things, For every value of x:

  • F(x) should be defined
  • And limit at x should be equal to f(x)

Examples:

Let

$$fx=\frac{x^4-1}{x-1}$$

for all real numbers

Continuity Function

Solutions:

The function at x=1 is not defined

$$fx=\frac{x^4-1}{x-1}=\frac{0}{0}$$

So,it is continuous function as point( 1) f(x) should be defined is not satisfied

let take x<1 then so now fx does not contain x=1 so now it is continuous at x<1 (does not have hole or break)

Piecewise Function

A type of function in which we often use more than one formula for defining the output and each output has its own domain

f(x)=formula 1 if x is in domain1 formula 2 if x is in domain 2

Example of Piecewise Function

g(x)={3 if x ≤ 1 x if x>

Continuity Function

Since, it is defined when x=1 as g(1)=3 (no break)

Also, at x=1 you can’t say like that what is limit of function as there are two points:

  • 3 from left side
  • 1 from right side

So,the limit does not exist at x=1 as there is a break in the graph so we can say that function is not continuous which means it is discontinuous.

Example:

The absolute value function in piecewise notation:

|x|=[x if x≥0 -x if x<0]

Continuity Function

Evaluation in Piecewise-Defined Function:

$$F(x)=8x+3$$

where x ≤ 0

=8x+6 where x ≥ 0

Evaluate the following:

  • F(-1)
  • F(3)

f(x) is defined on as 8x+3 for x=-1 as -1<0

$$f(-1)=8(-1)+3=-5$$

f(x) is defined on 8x+6 for x=3 as 3≥0

$$f(3)=8(3)+6$$

$$=30$$