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For each one that is not monotonic decreasing, name a subset of
the domain on which it is monotonic decreasing.
4 If
f
:
is monotonic increasing throughout its domain and the
set
V
f
(
x
)
x
R
is bounded, prove that lim
R
R
f
(
x
)
sup
V
.In
this casetheline
y
sup
V
is called an
asymptote
to thegraph of
thefunction.
5 Let
f
:
bea function which is monotonic increasing
throughout its domain, and let
a
bea point of thedomain.
Let
R
R
.
Find an upper bound for
L
and a lower bound for
U
.
Deduce that sup
L
and inf
U
exist.
Provethat sup
L
lim
L
f
(
x
)
x
a
and let
U
f
(
x
)
a
x
f
(
x
).
Thus one-sided limits exist for monotonic increasing functions
whether they are continuous or not. What are the values of these
limits for thefunction
x
[
x
] at integer points of the domain?
f
(
x
) and inf
U
lim
6 Formulate an analogue of qn 5 for monotonic decreasing functions,
and give an example of a discontinuous decreasing function and of
its one-sided limits at points of discontinuity.
7 If, in qn 5,
f
is continuous at
a
, show that sup
L
inf
U
.If
conversely, sup
inf
U
, show that
f
is continuous at
a
.If
sup
L
inf
U
, explain why sup
L
inf
U
, and deduce that a
rational number must lie between these two numbers. Deduce that
a monotonic function can haveat most a countablenumbr of
discontinuities.
L
Intervals
8 Find the range of each of the functions defined below. The domain
in each case is
R
.
(i)
f
(
x
)
1.
(ii)
f
(
x
)
sin
x
.
(iii)
f
(
x
)
arctan
x
.
1
1
x
(iv)
f
(
x
)
.
(v)
f
(
x
)
e
.
(vi)
f
(
x
)
x
.
(vii)
f
(
x
)
x
.
