Graphics Reference
In-Depth Information
7
Continuityand completeness
Functions on intervals
Concurrent reading:
Hart, Spivak ch. 7.
Further reading:
Rudin ch. 4.
Monotonic functions: one-sided limits
Increasing and decreasing functions, like increasing and decreasing
sequences, have interesting properties and it is useful to be able to
namesuch functions.
If
f
(
x
)
f
(
y
), when
x
y
, thefunction
f
is said to be
increasing
,or
monotonic increasing
.
If
f
(
x
)
f
(
y
), when
x
y
, thefunction
f
is said to be
strictly
increasing
,or
strictly monotonic increasing
.
1 Which of thefollowing functions with domain
R
aremonotonic
increasing:
(i)
x
x
; (ii)
x
x
; (iii)
x
[
x
]: (iv)
x
x
[
x
];
(v)
x
tan
x
,(
x
(
n
)
); (vi)
x
e
?
For each one that is
not
monotonic increasing, name a subset of
thedomain on which it
is
monotonic increasing.
2 Define what is meant by saying that a real function is
decreasing
or
monotonic decreasing
.
Also define what is meant by saying that a real function is
strictly
decreasing
or
strictly monotonic decreasing
.
3 Which of thefollowing functions with domain R aremonotonic
decreasing:
(i)
x
x
; (ii)
x
x
; (iii)
x
1/
x and
0
0; (iv)
x
e
?
