Graphics Reference
In-Depth Information
f
f
f
f
f
(
f
...
)(
x
)
(
x
)
(
x
)
...
(
x
)
is continuous at
x
a
.
26
The product rule
If real functions
f
and
g
havethesamedomain
A
, and areboth
continuous at
x
a
A
, useqn 3.54(vi), theproduct rul, to prove
that thereal function
f
·
g
defined by
f
(
x
) ·
g
(
x
)
(
f
·
g
)(
x
)
is also continuous at
x
a
.
27 Which of qns 21
—
26 are required, and in which order, to prove that
thefunctions
f
: R
R, given by
(i)
f
(
x
)
2
x
, (ii)
f
(
x
)
2
x
1, (iii)
f
(
x
)
x
,
x
2
x
(iv)
f
(
x
)
, (v)
f
(
x
)
1,
arecontinuous on
R
?
28 Proveby induction that thefunction
f
:
R
R
, given by
f
(
x
)
x
, where
n
N, is continuous on R.
29 Provethat thefunction
f
: R
R, given by
f
(
x
)
a
a
x
a
x
...
a
x
,
where the
a
are real numbers, is continuous on R.
This establishes that polynomial functions are continuous on R.
30 The acid test of understanding the language we have introduced
comes when deciding at what points the function given by
f
(
x
)
1/
x
is continuous. With reference to qn 3, determine the
maximum domain of definition of this real function. Use qns 3.65
and 3.66, the reciprocal rule, to prove that this function is
continuous at every point where it is defined.
31 Use qn 30 to extend the result of qn 29 to give a family of
functions each one of which is continuous on R
0
.
Continuityin less familiar settings
32 Useqn 3.54(ii), theabsolutevaluerul, to provethat thefunction
f
: R
R given by
f
(
x
)
x
is continuous on R. Sketch the graph
of
f
near the origin.
33 Sketch the graph of the real function given by
f
(
x
)
(
x
x
),
and provethat it is continuous on
R
.
