Riemann’s example of function f for which exists for all x, but is not differentiable when x is a rational number with even denominator.
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Riemann’s example of function
f
for which exists for all
x
, but is not differentiable when
x
is a rational number with even denominator.
Riemann’s example of function
f
for which exists for all
x
, but is not differentiable when
x
is a rational number with even denominator.
What does a derivative look like? Can we find a function that can’t be a derivative but which can be integrated?
Does a derivative have to be continuous?
If
F
is differentiable at
x
=
a,
can
F '
(
x
) be discontinuous at
x
=
a
?
If
F
is differentiable at
x
=
a,
can
F '
(
x
) be discontinuous at
x
=
a
?
Yes!
x
2 0,
x x
0, 0.
F
'
x
2 0,
2
x
x x
0, 0.
x
0.
F
'
x
2 0,
2
x
F
lim
h
0
F h
h
2 lim
h
0 lim
h
0
h h
0 .
x x
0, 0.
x
0.
F
'
x
2 0,
2
x
F
lim
h
0
F h
h
2 lim
h
0 lim
h
0
h h
0 .
x x
0, 0.
x
0.
lim
x
0
F
'
F
does not exist, but does exist (and equals 0).
How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal?
How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal?
No!
If lim
x
c
f
' and lim
x
c
f
' exist, then they must be equal and they must equal
f
' .
Mean Value Theorem:
f
'
f
' lim
x
c
lim
x
c
x x
c c
lim
x
c
f
' , lim
x
c
f
' ,
x
k
c c
k
x
If lim
x
c
f
' and lim
x
c
f
' exist, then they must be equal and they must equal
f
' .
Mean Value Theorem:
f
'
f
' lim
x
c
lim
x
c
x x
c c
lim
x
c
f
' , lim
x
c
f
' ,
x
k
c c
k
x
The derivative of a function cannot have any jump discontinuities!
Bernhard Riemann (1852, 1867)
On the representation of a function as a trigonometric series
a b
dx
x i
x i
1
Bernhard Riemann (1852, 1867)
On the representation of a function as a trigonometric series
a b
dx
x i
x i
1 Key to convergence: on each interval, look at the
variation
of the function
V i
sup
x
[
x i
1 ,
x i
] inf
x
[
x i
1 ,
x i
]
Bernhard Riemann (1852, 1867)
On the representation of a function as a trigonometric series
a b
dx
x i
x i
1 Key to convergence: on each interval, look at the
variation
of the function
V i
sup
x
[
x i
1 ,
x i
] inf
x
[
x i
1 ,
x i
] small as we wish by taking sufficiently small intervals.
Any continuous function is integrable: Can make
V i
as small as we want by taking sufficiently small intervals:
V i
x i
x i
1
b
a
x i
x i
1
b
a
b
a
.
Bernhard Riemann (1852, 1867)
On the representation of a function as a trigonometric series
Riemann gave an example of a function that has a jump discontinuity in
every
subinterval of [0,1], but which can be integrated over the interval [0,1].
Riemann’s function:
n
1
n
2
x
nearest integer , when this is 1 2, 0, when distance to nearest integer is 1 2 –2 –1 1
n
2 has
n
jumps of size 2
n
2 between 0 and 1 2
2 8
b
2
Riemann’s function:
n
1
n
2 2 8
b
2 The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number.
Riemann’s function:
n
1
n
2 2 8
b
2 The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number.
Conclusion : 0
x dt
exists and is well - defined for all
x
, but
F
is
not
differentiable at any rational number with an even denominator.