4. Continuity of Functions
Function
is continuous at point 
if
the following three conditions are satisfied :
i.) 
is defined ,
ii.)
exists (i.e., is finite) ,
iii.)
.
Function f is said to be continuous on an interval I if f is continuous at each point x in I. Here is a list of some well-known facts related to continuity :
1. The sum of continuous functions is continuous.
2. The difference of continuous functions is continuous.
3. The product of continuous functions is continuous.
4. The quotient of continuous functions is continuous at all points x where the denominator is not zero.
5. The functional composition of continuous functions is continuous at all points x where the composition is properly defined.
6. Any polynomial is continuous for all values of x.
7. Function ex and
trigonometry functions 
and 
are
continuous for all values of x.
f (x) is not continuous at x = a, then f (x) is said to be discontinuous at this point. Figure 1 shows the graphs of four functions, two of which are continuous at x = a and two are not.
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|
|
Continuous at x = a. |
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Discontinuous at x = a. |
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|
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Continuous at x = a. |
|
Discontinuous at x = a. |
|
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Classification of Discontinuity Points
All discontinuity points are divided into discontinuities of the first and second kind. The function f (x) has a discontinuity of the first kind at x = a if
There exist left-hand limit
and
right-hand limit 
;These one-sided limits are finite.
Further there may be the following two options:
The right-hand limit and the left-hand limit are equal to each other:
Such a point is called a removable discontinuity.
The right-hand limit and the left-hand limit are unequal:
In this case the function f (x) has a jump discontinuity.
The function f (x) is said to have a discontinuity of the second kind (or a non-removable or essential discontinuity) at x = a, if at least one of the one-sided limits either does not exist or is infinite.
Lecture 5 The derivative of the function. Geometric and mechanical meaning of derivative. Table of derivatives. The differential of a function
1. The derivative of a function
Concept of a derivative
Let
be a fixed point and
be a variable point on the curve
as shown on about figure. Then the slope of the line AP
is given by
or
.
When the variable point P
moves closer and closer to A
along the curve
,
i.e.
.
the line AP
becomes the tangent line of the curve at the point A.
Hence, the slope of the tangent line at the point A
is equal to
.
This term is defined to be the derivative of
at
and is usually denoted by
.
The definition of derivative at any point x may be defined as
follows.
Definition 1. Let
be a function defined on the interval
and
.
is said to be differentiable at
( or have a derivative at
) if the limit
exists. This lime value is denoted by
or
and is called the derivative of
at
.
If
has a derivative at every point x in
,
then
is said to be differentiable on
.
Remark. As
,
the difference between x and
is very small, i.e.
tends to zero.
Usually, this difference is denoted by h or
.
Then the derivative at
may be rewritten as
.
( First Principle )
Table of derivatives and basic rules differentiation
1.
2.
;
in particular
3.
;
in particular
4.
;
in particular
5.
6.
7.
8.
9.
10.
11.
12.
Basic rules for finding derivative If С is a constant and u, v are functions of х that have derivatives, then:
13.
14.
15.
16.
If
and
that is
where the functions y
and u have
derivatives, then
.
I. Notations for the Derivative
The derivative of
may be written in any of the following ways:
,
,
,
,
or
.
II. Basic Differentiation Rules
A. Suppose c and n are constants, and f and g are differentiable functions.
(1)
(2)
(3)
(Product Rule)
(4)
.
This derivative rule is called the Quotient Rule.
(5)
(6)
(7)
(Power Rule)
Differentiation of Composite Function
Suppose that there is a composite function denoted by the two-component chain y=F(u); u=(x), each of these functions is differentiable F(u), (x). Find the derivative of the composite function F[ (x)].
Theorem
1.
If functions F(u)and
(x)
are differentiable with respect to their variables, then there is a
derivative of composite function. It is defined as follows:
.
Derivative of an Inverse Function
Suppose that there is function y=f(x), being monotone, continuous and having one derivative yх . Suppose that there is the continuous derivative ху, of the inverse function x=f -1(у)=g(y)..
Theorem 2. If function y=f(x) is monotone and differentiable, then there is the derivative of inverse function. It is defined by the following formula:
or
.