Math 2413
Continuity
Observation:
Section 1.4 Notes
There is a hole on the graph of function f(x) at x = -2. We say this function is NOT continuous at x = -2.
There is a vertical asymptote x= 4. We say this function is NOT continuous at x = 4.
At x = 2, there is a gap between 2 pieces of a piece wise function. We say this function is NOT continuous at x
= 2.
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Math 2413
Definition:
Section 1.4 Notes
Let f be a function defined at least on an open interval (c – p, c + p) with p > 0. We say that f is continuous at c
if
lim f ( x )  f (c)
x c
According to the definition of limits, a function f is continuous at a point c if f is defined at x= c and
lim f ( x )  lim f ( x )  f (c )
x c 
x c
The function f is said to be discontinuous at c if it is not continuous there.
If the domain of f contains an interval (c – p, c + p), p > 0 (so that f is defined at c), then f can fail to be
continuous at c for only one of two reasons: either
i/ lim f ( x)  DNE
x c
ii/ lim f ( x)  f (c)
x c
Types of Discontinuity
Removable discontinuity
Jump discontinuity
f(c) and lim f ( x) exists but
Each one-sides limit exists but
Infinite discontinuity
f (x )   at least one side of c.
x c
lim f ( x)  f (c)
they are not equal
x c
This type is generally associated
with having a vertical
asymptote.
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Math 2413
Section 1.4 Notes
Remark: Recall that lim f ( x) exists if and only if one-sided limits exist and are equal. The existence of the
x c
limit does not depend on whether the function is defined at c or not. However, for continuity, this is a
requirement.
Geometrically speaking, a function is continuous at every number in an interval if the graph has no breaks in it.
That is, you can graph it without lifting your pen from the paper. When the graph is given, it is quite simple to
observe the points of discontinuity.
Fact: The following types of functions are continuous at every number in their domains:
• Polynomials
• Rational functions
• Root functions
• Trigonometric functions
• Inverse trigonometric functions
• Exponential functions
• Logarithmic functions
Many complicated continuous functions can be built using simple ones.
Theorem: If f and g are continuous at c, then
1) f + g is continuous at c
2) f – g is continuous at c
3) α f is continuous at c for each real number α
4) f × g is continuous at c
5) f / g is continuous at c provided g(c) ≠ 0
Theorem: If g is continuous at c and f is continuous at g(c), then the composition f ᵒ g is continuous at c.
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Math 2413
Definition: One-Sided Continuity
Section 1.4 Notes
A function f is called
Continuous from the left at c if lim f ( x)  f (c)

xc
And is called continuous from the right at c if lim f ( x )  f (c )

xc
f is continuous at c iff
lim f ( x ),
x c 
lim f ( x),
f (c )
x c 
all exist and equal
Remark:
The polynomial functions are continuous everywhere.
The rational functions are continuous everywhere on its domain.
Example 1: Find all values of x so that each function f is discontinuous and classify the continuity at each value
of x.
Continuity over an interval
Definition: Let (a, b) be an open interval. A function is said to be continuous on the interval (a, b) if it is
continuous at every number in this interval.
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Math 2413
Section 1.4 Notes
If f is defined on a closed interval [a, b], we only expect to have one-sided continuity at the end points a and b.
That is, if the function is continuous at every number in (a, b), continuous from the right at a and continuous
from the left at b, then we say that the function is continuous on the interval [a, b].
These definitions can be extended to functions defined on half-open intervals or infinite intervals.
Examples: Find all values of x so that each function f is discontinuous then find all intervals for which f is
continuous.
1.
x3  2 x  4
f ( x) 
x2  1
2.
x2  2x  5
f ( x) 
x4
3.
f ( x) 
x2  9
x3
5
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Math 2413
Section 1.4 Notes
4.
f ( x)  x  5
5.
f ( x) 
x 2
x  4x  4
6.
f ( x) 
2x  5x2
x
2
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Math 2413
Section 1.4 Notes
7.
 3x; x  1
f ( x)  
x  2; x  1
8.
 x2; x  0
f ( x)  
1  x; x  0
9.
 2x2  9

f ( x)   2
 x3

x3
x3
3 x
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Math 2413
Section 1.4 Notes
2
 x

3
10. f ( x )  
2-x
1
 2
x
x  -1
x  -1
-1  x  1
1 x
Examples: Find c so that F(x) is continuous.
1.
2 x  3;
f ( x)  
 cx - x
x2
x2
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Math 2413
2.
 x  2;
f ( x)  
cx - 3
Section 1.4 Notes
x5
x5
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