define continuity of a function
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Answer:
Definition of Continuity
A function f(x) is said to be continuous at a point x = a, in its domain if the following three conditions are satisfied: f(a) exists (i.e. the value of f(a) is finite) Limx→a f(x) exists (i.e. the right-hand limit = left-hand limit, and both are finite)
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Answer:
A real function
f
(
x
)
is said to be continuous at
a
∈
R
(
R
−
is the set of real numbers), if for any sequence
{
x
n
}
such that
lim
n
→
∞
x
n
=
a
,
it holds that
lim
n
→
∞
f
(
x
n
)
=
f
(
a
)
.
In practice, it is convenient to use the following three conditions of continuity of a function
f
(
x
)
at point
x
=
a
:
Function
f
(
x
)
is defined at
x
=
a
;
Limit
lim
x
→
a
f
(
x
)
exists;
It holds that
lim
x
→
a
f
(
x
)
=
f
(
a
)
.
Cauchy Definition of Continuity
(
ε
−
δ
−
Definition)
Consider a function
f
(
x
)
that maps a set
R
of real numbers to another set
B
of real numbers. The function
f
(
x
)
is said to be continuous at
a
∈
R
if for any number
ε
>
0
there exists some number
δ
>
0
such that for all
x
∈
R
with
|
x
−
a
|
<
δ
,
the value of
f
(
x
)
satisfies:
|
f
(
x
)
−
f
(
a
)
|
<
ε
.
Definition of Continuity in Terms of Differences of Independent Variable and Function
We can also define continuity using differences of independent variable and function. The function
f
(
x
)
is said to be continuous at the point
x
=
a
if the following is valid:
lim
Δ
x
→
0
Δ
y
=
lim
Δ
x
→
0
[
f
(
a
+
Δ
x
)
−
f
(
a
)
]
=
0
,
where
Δ
x
=
x
−
a
.
All the definitions of continuity given above are equivalent on the set of real numbers.
A function
f
(
x
)
is continuous on a given interval, if it is continuous at every point of the interval.
Continuity Theorems
Theorem
1.
Let the function
f
(
x
)
be continuous at
x
=
a
and let
C
be a constant. Then the function
C
f
(
x
)
is also continuous at
x
=
a
.
Theorem
2.
Let the functions
f
(
x
)
and
g
(
x
)
be continuous at
x
=
a
. Then the sum of the functions
f
(
x
)
+
g
(
x
)
is also continuous at
x
=
a
.
Theorem
3.
Let the functions
f
(
x
)
and
g
(
x
)
be continuous at
x
=
a
.
Then the product of the functions
f
(
x
)
g
(
x
)
is also continuous at
x
=
a
.
Theorem
4.
Let the functions
f
(
x
)
and
g
(
x
)
be continuous at
x
=
a
. Then the quotient of the functions
f
(
x
)
g
(
x
)
is also continuous at
x
=
a
assuming that
g
(
a
)
≠
0
.
Theorem
5.
Let
f
(
x
)
be differentiable at the point
x
=
a
.
Then the function
f
(
x
)
is continuous at that point.
Remark: The converse of the theorem is not true, that is, a function that is continuous at a point is not necessarily differentiable at that point.
Theorem
6
(Extreme Value Theorem).
If
f
(
x
)
is continuous on the closed, bounded interval
[
a
,
b
]
, then it is bounded above and below in that interval. That is, there exist numbers
m
and
M
such that
m
≤
f
(
x
)
≤
M
for every
x
in
[
a
,
b
]
(see Figure
1
).
extreme-value-theorem
Figure 1.
Theorem
7
(Intermediate Value Theorem).
Let
f
(
x
)
be continuous on the closed, bounded interval
[
a
,
b
]
. Then if
c
is any number between
f
(
a
)
and
f
(
b
)
, there is a number
x
0
such that
f
(
x
0
)
=
c
.
The intermediate value theorem is illustrated in Figure
2.
intermediate-value-theorem
Figure 2.
Continuity of Elementary Functions
All elementary functions are continuous at any point where they are defined.
An elementary function is a function built from a finite number of compositions and combinations using the four operations (addition, subtraction, multiplication, and division) over basic elementary functions. The set of basic elementary functions includes:
Algebraical polynomials
A
x
n
+
B
x
n
−
1
+
…
+
K
x
+
L
;
Rational fractions
A
x
n
+
B
x
n
−
1
+
…
+
K
x
+
L
M
x
m
+
N
x
m
−
1
+
…
+
T
x
+
U
;
Power functions
x
p
;
Exponential functions
a
x
;
Logarithmic functions
log
a
x
;
Trigonometric functions
sin
x
,
cos
x
,
tan
x
,
cot
x
,
sec
x
,
csc
x
;
Inverse trigonometric functions
arcsin
x
,
arccos
x
,
arctan
x
,
arccot
x
,
arcsec
x
,
arccsc
x
;
Hyperbolic functions
sinh
x
,
cosh
x
,
tanh
x
,
coth
x
,
sech
x
,
csch
x
;
Inverse hyperbolic functions
arcsinh
x
,
arccosh
x
,
arctanh
x
,
arccoth
x
,
arcsech
x
,
arccsch
x
.