MATHEMATICS  THEORY





In this section, the concept of continuous and discontinuous
is introduced. Various theorems related to continuous function will be
discussed. 


Continuous

Continuous Function Example 

If
then function f(x) is continuous at a.
A continuous function implies that:
 f(a) is defined in the domain of f(x).
 The limit of f(x) exists at point a.
 The limit of f(x) at point a equals the value of function f(x) at point
a.

Discontinuous Function Example




Discontinuous


If f(x) is not continuous at point
a, then f(x) is discontinuous at a. In other words, f(x) has a discontinuity at point a. 





Theorems about Continuous

In this segment, some theorems are introduced to justify whether
a function is continuous or not.
 A function f(x) is continuous from the left at point a, if
 A function f(x) is continuous from the right at point a, if
 Assume that f(x) and g(x) are continuous at point a, then:
 The constant function h(x) = c is continuous for all x at every
a.
 cf(x) is continuous at a for any constant c.
 f(x) + g(x) is continuous at a.
 f(x)  g(x) is continuous at a.
 f(x)g(x) is continuous at a.
 f(x)/g(x) is continuous at a if g(a) ≠ 0.
 f^{1/n} is continuous at a if f(a)^{1/n }exists.
 Every polynomial function
is continuous.
 Every rational function
is continuous where
 f(x) and g(x) are polynomial functions.
 g(x) ≠ 0
 If f(x) is continuous at b and
then
 If g(x) is continuous at a and f(x) is continuous at g(a), then
is continuous at a. In other words, a continuous function of a continuous function
is a continuous function.






Intermediate Value Theorem

Explanation of
Intermediate Value Theorem< 

In order to justify whether a value will fall in a specific range
or not, intermediate value theorem is introduced.
Assume:
 f(x) is continuous on the interval [a, b]
 f(a) ≠ f(b)
 N is any number between f(a) and f(b)
then there exists a point c in [a, b] such that f(c) = N. 





Corollary of Intermediate Value Theorem

f(x) = 0 has 5 roots in [a, b] 

Corollary of Intermediate Value Theorem is a theorem used to
verify whether a function has at lease a root or not.
Assume:
 f(x) is continuous on the [a,b]
 f(a) and f(b) has opposite signs
then the equation f(x) = 0 has at least one root in the open interval
(a, b). 





Extreme Value Theorem

Continuous function
with minimum and maximum value 

This is a theorem that can be used for one
variable.
If f(x) is continuous on [a, b], then f(x) takes on a least
value m and a greatest value M on the interval.
Discontinuous function
without maximum value






Theorem

Theorem example 

If f(x) is continuous at point c and
f(c) > 0
then there is a
positive number t such that, whenever
c  t < x < c + t
then
f(x) > 0. 


