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 sin 1 , x  0,
x
Fx   
x  0.
 0,
 
x 2 sin 1 , x  0,
x
Fx   
x  0.
 0,
 
 
F' x   2xsin 1 x  cos 1 x , x  0.
 
x 2 sin 1 , x  0,
x
Fx   
x  0.
 0,
 
 
F' x   2xsin 1 x  cos 1 x , x  0.
F h   F 0 
F' 0  lim
h 0
h
h 2 sin 1 h
 lim
h0
h
 lim h sin 1 h  0 .
h0
 
 
 
x 2 sin 1 , x  0,
x
Fx   
x  0.
 0,
 
 
F' x   2xsin 1 x  cos 1 x , x  0.
F h   F 0 
F' 0  lim
h 0
h
lim F' x  does not exist, but
2
1
x0
h sin h
 lim
F' 0 does exist (and equals 0).
h0
h
 
 
 lim h sin 1 h  0 .
h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 f ' x  and lim f' x exist,
xc
xc
then they must be equal and they
must equal f ' c.
Mean Value Theorem:
f  x  f c
f ' c  lim
 lim f ' k, x  k  c
x c
x c
x c
f x  f c
f ' c  lim
 lim f ' k, c  k  x
x c
xc
x c
If lim f ' x  and lim f' x exist,
xc
xc
then they must be equal and they
must equal f ' c.
Mean Value Theorem:
f  x  f c
f ' c  lim
 lim f ' k, x  k  c
x c
x c
x c
f x  f c
f ' c  lim
 lim f ' k, c  k  x
x c
xc
x c
The derivative of a function cannot
have any jump discontinuities!
Bernhard Riemann (1852, 1867) On the representation
of a function as a trigonometric series

b
Defined a

f
x

f  xdx as limit of  i xi  xi1 
Bernhard Riemann (1852, 1867) On the representation
of a function as a trigonometric series

b
Defined a

f
x

f  xdx as limit of  i xi  xi1 
Key to convergence: on each interval, look at the
variation of the function
Vi 
sup
x[ xi1, xi ]
f x  
inf
x[x i1 ,x i ]
f x
Bernhard Riemann (1852, 1867) On the representation
of a function as a trigonometric series

b
Defined a

f
x

f  xdx as limit of  i xi  xi1 
Key to convergence: on each interval, look at the
variation of the function
Vi 
sup
x[ xi1, xi ]
f x  
inf
x[x i1 ,x i ]
f x
Integral exists if and only if  Vi xi  xi1  can be made as
small as we wish by taking sufficiently small intervals.
Any continuous function is integrable:
Can make Vi as small as we want by taking
sufficiently small intervals:
 Vi xi  xi 1  

ba
 xi  xi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:
f x  


n1
n x
n2

x  nearest integer , when this is  1 2 ,
x  
1
0,
when
distance
to
nearest
integer
is


2
–2
–1
n x
n2
1
2
has n jumps of size 2 between 0 and 1
n
2
2

a
At x  2b , gcd a,2b  1, the function jumps by
8b 2
Riemann’s function:
f x  


n1
n x
n2
2

a
At x  2b , gcd a,2b  1, the function jumps by
8b 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:
f x  


n1
n x
n2
2

a
At x  2b , gcd a,2b  1, the function jumps by
8b 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: Fx  
0 f t  dt exists and is well - defined
x
for all x, but F is not differentiable at any
rational number with an even denominator.