Calculus Section 1.4 Notes and Examples – “Continuity and One Sided Limits”
As we have encountered in the past, we know that some functions are continuous over the entire domain of real
numbers and some are not. Examples of functions that are continuous over the entire set of Real numbers
would be the following functions:


Polynomial Functions
y=sinx and y =cosx

y  ax
( For a  0, a  1)
These are the best-behaved functions we have. There are still other functions that are not continuous over the
entire domain of real numbers, but they are continuous over their implied domains. Example of functions that
meet this condition would be:

Radical functions (Like y 
n
x

y  tan x, y  cot x, y  sec x, y  csc x

y  logb x
When a function is not continuous at some real number “c” , the behavior at “c” can be varied. Below are some
different behaviors that we have already witnessed at points of discontinuity.
c
Discontinuous at “c”
f (c) is not defined
Limit does exist at x = c
c
Discontinuous at “c”
f (c) is defined
Limit does not exist at x = c
c
Discontinuous at “c”
f (c) is defined
Limit does exist but lim f ( x)  f (c)
x c
Conditions for Continuity at a Point
In order for a function to be considered continuous at a point “c” 3 conditions must first be met.
Condition 1: f (c ) must be defined. In graph 1 above that condition is not met
Condition 2: The limit as “x” approaches “c” must exist. In graph 2 above that condition is violated
Condition 3: The limit of f ( x ) at x = c must equal f (c). In graph 3 above that condition is violated.
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Formal Definition of Continuity on an open interval.
A function is continuous on an open interval (a, b) if it is continuous at each point from a to b. A
function that is continuous on the entire real number (polynomial function, sine function, cosine,
function, power function, log functions,) is said to be everywhere continuous.
Discontinuities of Functions can be described as either removable or non-removable. Suppose over an open
interval a function is defined at every “x” value except for possibly x = c. If the function f is not continuous at “c”
then “c” can be considered either a removable or non-removable continuity:
Removable discontinuity – A removable discontinuity at “c” is a location on the function that can easily be filled
with one point. (Fill the hole and the graph is continuous) A Limit exists at these discontinuities.
Non-removable Discontinuity – a non-removable discontinuity at “c” is a location where there is jump break in
the graph or a big gap before and after “c”. (No limits where there are non-removable discontinuities)
c
removable discontinuity at c
c
removable discontinuity at c
c
non-removable discontinuity at c
More Functions that have discontinuities
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 x 1 
f ( x)  
 A nonremovable discontinuity at x = 1
 x 1 
f ( x)  x
Discontinuous at any integer value
Discontinuity is – removable
f ( x) 
x2  x  2
x2
A rational function discontinuous at x=2
But this is a removable discontinuity since there
Is only a single hole which can be filled to make
the function continuous. Again this function is
continuous over its domain but not everywhere continous
 x2 ,

f ( x)  6,
10  x,

x2
x2
x2
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A Piecewise Function with a non-removable discontinuity
at x = 2.
Continuity on a Closed Interval
To understand continuity on a closed interval you must first understand a one-sided limit. A one sided limit can
either be determined from approaching some target value “c” from the right. The way this is notated is as
follows:
Right sided limit
lim f ( x) This is read, the limit of f(x) as you approach "c" from the right
x c
Left sided limit
lim f ( x) This is read, the limit of f(x) as you approach "c" from the left
x c 
Examples of one sided limits
lim f ( x ) 
x 1
lim f ( x ) 
x 1
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Recall that in a function, a limit only exists at an “x” value if the right hand behavior and the left behavior are the
same as you approach that “x”. In the function above this is not the case as “x” approaches 1 so this function
would not have a limit as “x” approaches 1.
So what is required to have continuity over a closed interval?
a
b
Figure 1
a
b
Figure 2
The figure 1 above is continuous over the open interval (a, b) But not on the closed interval  a, b because of
the holes at the end where “a” and “b” are located. If however those two holes were filled in at “a” and “b”,
like in figure 2, then there would be both continuity over the open interval (a, b) as well as continuity over the
closed interval  a, b .
Requirement for a Function to be continuous over a closed interval. (Exam the endpoints of the interval)
A function is said to be continuous over a closed interval if it is continuous over the open interval and the right
hand limit of f ( x ) at “a” is f ( a ) and the left hand limit of f ( x ) at “b” is f (b ) .
Recall that the following limit properties apply to the limits of two different functions.
lim af ( x )  a the limit of f ( x )
x c
lim  f ( x )  g ( x )   the sum or difference of the limits of f  x  and g  x 
x c
lim  f ( x )  g ( x )   The product of the limits of f  x  and g  x 
x c
 f ( x) 
lim 
  The quotient of the limits of f  x  and g  x 
x c
 g ( x) 
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Similarly If two different functions are continuous at some point “c” the following continuity properties would
apply.
If f ( x)  is continuous at "c" then af ( x)is continuous at "c"
If f ( x) and g  x  are continuous at "c" then the sum or differenceof f  x  and g  x  is continuous at "c"
If f ( x) and g  x  are continuous at "c" The product of f  x  and g  x  is continuous at "c"
If f ( x) and g  x  are continuous at "c"The quotient f  x  and g  x  is continuous at "c"
If g ( x) is continuous at "c" and f ( x) is continuous at g (c), then  f g  (c) is continous at "c".
The continuity of a function leads to a very useful mechanism in math which is known as
the intermediate value theorem.
The intermediate value theorem states the following:
If f is a continuous function on the closed interval  a, b  and "k" is any real number
between f (a) and f  b  then there is at least one value "c" for which f (c)  k .
k
a
c
b
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