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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!
F x( ) =x2 sin 1
x( ), x≠0,0, x=0.
⎧ ⎨ ⎩
F x( ) =x2 sin 1
x( ), x≠0,0, x=0.
⎧ ⎨ ⎩
F' x( ) =2xsin 1x( )−cos1x( ), x≠0.
F x( ) =x2 sin 1
x( ), x≠0,0, x=0.
⎧ ⎨ ⎩
F' x( ) =2xsin 1x( )−cos1x( ), x≠0.
F ' 0( ) =limh→ 0
F h( )−F 0( )h
=limh→ 0
h2 sin 1h( )
h
=limh→ 0
hsin 1h( ) =0 .
F x( ) =x2 sin 1
x( ), x≠0,0, x=0.
⎧ ⎨ ⎩
F' x( ) =2xsin 1x( )−cos1x( ), x≠0.
F ' 0( ) =limh→ 0
F h( )−F 0( )h
=limh→ 0
h2 sin 1h( )
h
=limh→ 0
hsin 1h( ) =0 .
limx→ 0
F' x( ) ,does not exist but
F' 0( ) ( 0).does exist and equals
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 limx→ c−
f' x( ) andlimx→ c+
f' x( ) ,exist
then they must be equal and they
must equalf' c( ).Mean Value Theorem:
f ' c( ) = limx→ c−
f x( ) −f c( )x−c
= limx→ c−
f' k( ), x< k< c
f' c( ) = limx→ c+
f x( ) −f c( )x−c
= limx→ c+
f' k( ), c < k< x
The derivative of a function cannot have any jump discontinuities!
If limx→ c−
f' x( ) andlimx→ c+
f' x( ) ,exist
then they must be equal and they
must equalf' c( ).Mean Value Theorem:
f ' c( ) = limx→ c−
f x( ) −f c( )x−c
= limx→ c−
f' k( ), x< k< c
f' c( ) = limx→ c+
f x( ) −f c( )x−c
= limx→ c+
f' k( ), c < k< x
Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series
Defined as limit of f xi∗( )∑ xi −xi−1( )f x( )
a
b∫ dx
Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series
Defined as limit of f xi∗( )∑ xi −xi−1( )
Key to convergence: on each interval, look at the variation of the function
Vi = supx∈[ xi−1,xi ]
f x( ) − infx∈[xi−1 ,xi ]
f x( )
f x( )a
b∫ dx
Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series
Defined as limit of f xi∗( )∑ xi −xi−1( )
Key to convergence: on each interval, look at the variation of the function
f x( )a
b∫ dx
Vi = supx∈[ xi−1,xi ]
f x( ) − infx∈[xi−1 ,xi ]
f x( )
Integral exists if and only if can be made as small as we wish by taking sufficiently small intervals.
Vi∑ xi −xi−1( )
Any continuous function is integrable:
Can make Vi as small as we want by taking
sufficiently small intervals:
Vi∑ xi −xi−1( ) <ε
b−axi −xi−1( )∑ = ε
b−ab−a( ) =ε.
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:
x{ } =x− nearest integer( ), when this is< 1
2,0, when distance to nearest integer is1
2
⎧ ⎨ ⎪
⎩ ⎪
1 2–1–2
n x{ }n2
hasn jumps of size2n2 0 1between and
f x( ) =nx{ }n2n=1
∞
∑
At the function jumps by x =a2b, gcda,2b( ) =1, π 2
8b2
Riemann’s function: f x( ) =nx{ }n2n=1
∞
∑
At the function jumps by x =a2b, gcda,2b( ) =1, π 2
8b2
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( ) =nx{ }n2n=1
∞
∑
At the function jumps by x =a2b, gcda,2b( ) =1, π 2
8b2
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( ) = f t( )0
x∫ dt exists and is well- defined
for allx, butF isnot differentiable at any
.rational number with an even denominator