Graphics Reference
In-Depth Information
18 Describe the subset of
of values of
x
at which the integer
function of qn 12 is
not
continuous. Also describe the subset of R of
values of
x
at which this function
is
continuous.
R
The ideas in questions 12
—
18 lead to the following definition.
The function
f: A
R is continuous at
a
A
R when, for every
sequence (
a
n
) converging to
a
, with terms in
A
, the sequence (
f
(
a
n
))
converges to
f
(
a
).
This is commonly expressed by saying that '
f
(
x
) tends to
f
(
a
)as
x
tends to
a
' which is also expressed symbolically by writing
f
(
x
)
f
(
a
)as
x
a
.
Thefunction
f
:
A
R is said to be
continuous on A
, when it is
continuous at every
a
A
.
Examples of discontinuity
Continuous functions areso familiar that to clarify themeaning of
this definition we need some examples of discontinuity, illustrating the
absence of continuity. The continuity of a function
f
at a point
a
is
recognised by the way the function affects
every
sequence (
a
) tending to
a
. Discontinuity is established by finding just one sequence (
a
)
a
for
which (
f
(
a
)) does not tend to
f
(
a
).
19 Examinethegraph of
y
sin 1/
x
, paying particular attention to the
values of
y
when
x
is small.
Wedefinea function
f
: R
R by
f
(
x
)
sin 1/
x
when
x
0, and
f
(0)
0.
Let
a
1/(2
n
) and
b
1/((2
n
)
).
