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