
Differential
calculus 


Properties
of continuous functions 
Continuous function definition 
Intermediate value theorem  Bolzano's theorem 
Existence of roots 
Extreme value theorem 
Monotone function 





Properties
of continuous functions 
Continuous
function definition 
A
real function
y = f (x)
is continuous at a point a
if it is defined at x =
a
and 

that
is, if for every e
> 0 there is a d(e)
> 0 such that 
f
(x)

f
(a)
 < e
whenever  x

a  < d(e). 
Therefore,
if a function changes gradually as independent variable changes,
so that at every value a,
of the independent
variable, the difference between f
(x)
and f
(a)
approaches zero as x
approaches a. 
Thus,
a function is continuous at a point if both onesided limits (the
lefthanded and righthanded limits) are equal, 

that
is, if it is continuous both on the left and on the right at that point. 
A
point at which the function value is not equal to its limit, as x
approaches that point, is called point
of discontinuity.
A
function having points of discontinuity is discontinuous. 
A
function is said to be continuous if it is continuous at all
points 

1.
Intermediate value theorem  Bolzano's theorem 
If
a function is continuous on the closed interval [a,
b] then it takes every value
between f (a)
and f (b)
for at least
one argument between a
and b. 
That
is, for every y
between f
(a)
and f (b)
there exists at least
one argument x
between a
and b whose
function's
value f (x)
= y, as
shows the figure below. 


2.
Existence of roots 
If
a function is continuous on the closed interval [a,
b] then, if f
(a)
and f (b)
have opposite signs, or 
f
(a)
·
f (b)
< 0, then there exists x
Î
[a, b] such that f
(x) = 0. 
That
is, the function has at least one real root. For example,
inside the closed interval [a, b],
the function shown
in the figure below, has three roots, x_{1},
x_{2}
and x_{3},

that is, f
(x_{1}) = 0, f
(x_{2}) = 0
and f
(x_{3}) = 0 since
f(a)
·
f(b) < 0. 


3.
Extreme value theorem 
If
a function is continuous on the closed interval [a,
b] then, there
exist x_{min}
and x_{Max}
Î
[a, b] such that 
for all x
Î
[a, b], values of the function
f
(x_{min})
< f
(x) < f
(x_{Max}). 


4.
Monotone function 
A continuous
function is monotone on an interval if it is consistently
increasing or decreasing in value,

so that
either
f
(x_{1}) < f
(x_{2})
for
all
x_{1} < x_{2},

or
f
(x_{1}) > f
(x_{2})
for
all
x_{1} < x_{2}.

These
may be called strictly monotone
functions to distinguish them from those satisfying either

f
(x_{1}) < f
(x_{2})
for
all
x_{1} < x_{2},

or
f
(x_{1}) > f
(x_{2})
for
all
x_{1} < x_{2},

that
are called weakly monotone. 








Functions
contents E 



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