Identify all discontinuities for the following functions as either a jump or a removable discontinuity.
Notice that the function is defined everywhere except at
Thus,
does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as
approaches 5 is 8, and Condition 2 is not satisfied.This means there is a removable discontinuity at
Condition 2 is satisfied because
Notice that the function is a
piecewise function , and for each piece, the function is defined everywhere on its domain. Let’s examine Condition 1 by determining the left- and right-hand limits as
approaches 2.
Left-hand limit:
The left-hand limit exists.
Right-hand limit:
The right-hand limit exists. But
So,
does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at
Recognizing continuous and discontinuous real-number functions
Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of
as
approaches
is the same as the value of
when
So
There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.
Examples of continuous functions
The following functions are continuous everywhere:
Polynomial functions
Ex:
Exponential functions
Ex:
Sine functions
Ex:
Cosine functions
Ex:
The following functions are continuous everywhere they are defined on their domain:
Logarithmic functions
Ex:
,
Tangent functions
Ex:
is an integer
Rational functions
Ex:
Given a function
determine if the function is continuous at
Check Condition 1:
exists.
Check Condition 2:
exists at
Check Condition 3:
If all three conditions are satisfied, the function is continuous at
If any one of the conditions is not satisfied, the function is not continuous at
Determining whether a piecewise function is continuous at a given number
Determine whether the function
is continuous at
To determine if the function
is continuous at
we will determine if the three conditions of continuity are satisfied at
.
Condition 1: Does
exist?
Condition 2: Does
exist?
To the left of
to the right of
We need to evaluate the left- and right-hand limits as
approaches 1.
Left-hand limit:
Right-hand limit:
Because
does not exist.
There is no need to proceed further. Condition 2 fails at
If any of the conditions of continuity are not satisfied at
the function
is not continuous at
Condition 1: Does
exist?
Condition 2: Does
exist?
To the left of
to the right of
We need to evaluate the left- and right-hand limits as
approaches
Left-hand limit:
Right-hand limit:
Because
exists,
Condition 3: Is
Because all three conditions of continuity are satisfied at
the function
is continuous at
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc