Graphics Reference
In-Depth Information
Summary
-
continuity by sequences
The sequence definition of continuity
qn 18
A function f : A R is said to be continuous at
a
A when, for every sequence ( a
)
a with
)) f ( a ).
Theorem If thefunctions f : A R and g : A R areboth
continuous at a A , then
terms in A ,( f ( a
qn 41
(i) f is continuous at a ;
qn 52
(ii) 1/ f is continuous at a , provided f ( a )
0;
qn 23
(iii) f g is continuous at a ;
qn 26
(iv) f ยท g is continuous at a ;
qn 54
(v) f / g is continuous at a , provided g ( a ) 0.
Definition
qn 38
If thefunction f is defined on the range of the
function g , then the function of f g is defined by
f g ( x ) f ( g ( x ))
If thefunction g is continuous at thepoint a ,
and thefunction f is continuous at thepoint
g ( a ), then the composite function f g is
continuous at thepoint a .
A squeeze rule If for three functions f , g and h
qn 36
Theorem
qn 40
(i) f ( x ) g ( x ) h ( x ) in a neighbourhood of the
point a ;
(ii) f ( a ) h ( a );
(iii) f and h arecontinuous at thepoint a ;
then the function g is continuous at a .
Theorem
qn 21
A constant function is continuous at each point
of its domain.
Theorem
qn 22
The identity function is continuous at each
point of its domain.
Theorem
qn 29
A polynomial function is continuous at each
point of its domain.
Neighbourhoods
Wesay that a function f is continuous at a point x a when ' f ( x )
tends to f ( a ), as x tends to a ', which wemay think of as meaning 'as x
gets near to a , f ( x ) gets near to f ( a )'. So far, we have always described
nearness using sequences and convergence. Another way to describe
nearness is to consider neighbourhoods of a point a .
56 Illustratethests
x :
1
x
3
,
x :
2
x
1
2
,
x :
x
1
2
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