Tag Archives: real-analysis

A discontinuous real convex function

Consider a function f defined on a real interval IR. f is called convex if: x,yI λ[0,1]: f((1λ)x+λy)(1λ)f(x)+λf(y)

Suppose that I is a closed interval: I=[a,b] with a<b. For a<s<t<u<b one can prove that: f(t)f(s)tsf(u)f(s)usf(u)f(t)ut. It follows from those relations that f has left-hand and right-hand derivatives at each point of the interior of I. And therefore that f is continuous at each point of the interior of I.
Is a convex function defined on an interval I continuous at all points of the interval? That might not be the case and a simple example is the function: f:[0,1]Rx0 for x(0,1)x1 else

It can be easily verified that f is convex. However, f is not continuous at 0 and 1.

Counterexamples around differentiation of sequences of functions

We consider here sequences of real functions defined on a closed interval. Following theorem is the main one regarding the differentiation of the limit.

Theorem: Suppose (fn) is a sequence of functions, differentiable on [a,b] and such that (fn(x0)) converges for some point x0[a,b]. If (fn) converges uniformly on [a,b], then (fn) converges uniformly on [a,b] to a function f and for all x[a,b] f(x)=limnfn(x) What happens if we drop some hypothesis of the theorem? Continue reading Counterexamples around differentiation of sequences of functions

Pointwise convergence and properties of the limit (part 1)

We look here at the continuity of a sequence of functions that converges pointwise and give some counterexamples of what happens versus uniform convergence.

Recalling the definition of pointwise convergence

We consider here real functions defined on a closed interval [a,b]. A sequence of functions (fn) defined on [a,b] converges pointwise to the function f if and only if for all x[a,b] limn+fn(x)=f(x). Pointwise convergence is weaker than uniform convergence.

Pointwise convergence does not, in general, preserve continuity

Suppose that fn : [0,1]R is defined by fn(x)=xn. For 0x<1 then limn+xn=0, while if x=1 then limn+xn=1. Hence the sequence fn converges to the function equal to 0 for 0x<1 and to 1 for x=1. Although each fn is a continuous function of [0,1], their pointwise limit is not. f is discontinuous at 1. We notice that (fn) doesn't converge uniformly to f as for all nN, supx[0,1]|fn(x)f(x)|=1. That's reassuring as uniform convergence of a sequence of continuous functions implies that the limit is continuous! Continue reading Pointwise convergence and properties of the limit (part 1)

A nowhere locally bounded function

In that article, I described some properties of Thomae’s functionf. Namely:

  • The function is discontinuous on Q.
  • Continuous on RQ.
  • Its right-sided and left-sided limits vanish at all points.

Let’s modify f to get function g defined as follow:
g:|RRx0 if xRQpqq if pq in lowest terms and q>0 f and g both vanish on the set of irrational numbers, while on the set of rational numbers, g is equal to the reciprocal of f. We now consider an open subset OR and xO. As f right-sided and left-sided limits vanish at all points, we have limn+f(xn)=0 for all sequence (xn) of rational numbers converging to x (and such a sequence exists as the rational numbers are everywhere dense in the reals). Hence limn+g(xn)=+ as f is positive.

We can conclude that g is nowhere locally bounded. The picture of the article is a plot of function g on the rational numbers r=pq in lowest terms for 0<r<1 and q50.

A function continuous at all irrationals and discontinuous at all rationals

Let’s discover the beauties of Thomae’s function also named the popcorn function, the raindrop function or the modified Dirichlet function.

Thomae’s function is a real-valued function defined as:
f:|RRx0 if xRQpq1q if pq in lowest terms and q>0

f is periodic with period 1

This is easy to prove as for xRQ we also have x+1RQ and therefore f(x+1)=f(x)=0. While for y=pqQ in lowest terms, y+1=p+qq is also in lowest terms, hence f(y+1)=f(y)=1q. Continue reading A function continuous at all irrationals and discontinuous at all rationals

A decreasing function converging to zero whose derivative diverges (part2)

In that article, I gave examples of real valued functions defined on (0,+) that converge to zero and whose derivatives diverge. But those functions were not monotonic. Here I give an example of a decreasing real valued function g converging to zero at + and whose derivative is unbounded.

We first consider the polynomial map:
P(x)=(1+2x)(1x)2=13x2+2x3 on the segment I=[0,1]. P derivative equals P(x)=6x(1x). Therefore P is decreasing on I. Moreover we have P(0)=1, P(1)=P(0)=P(1)=0 and P(1/2)=3/2. Continue reading A decreasing function converging to zero whose derivative diverges (part2)

Differentiable functions converging to zero whose derivatives diverge (part1)

In this article, I consider real valued functions f defined on (0,+) that converge to zero, i.e.:
limx+f(x)=0 If f is differentiable what can be the behavior of its derivative as x approaches +?

Let’s consider a first example:
f1:(0,+)Rx1x f1 derivative is f1(x)=1x2 and we also have limx+f1(x)=0. Let’s consider more sophisticated cases! Continue reading Differentiable functions converging to zero whose derivatives diverge (part1)

A continuous function not differentiable at the rationals but differentiable elsewhere

We build here a continuous function of one real variable whose derivative exists on RQ and doesn’t have a left or right derivative on each point of Q.

As Q is (infinitely) countable, we can find a bijection nrn from N to Q. We now reuse the function f defined here. Recall f main properties: Continue reading A continuous function not differentiable at the rationals but differentiable elsewhere

A differentiable function except at one point with a bounded derivative

We build here a continuous function of one real variable whose derivative exists except at 0 and is bounded on R.

We start with the even and piecewise linear function g defined on [0,+) with following values:
g(x)={0if x=00if x{k4n;(k,n){1,2,4}×N}1if x{34n;nN}
The picture below gives an idea of the graph of g for positive values. Continue reading A differentiable function except at one point with a bounded derivative

A continuous function which is not of bounded variation

Introduction on total variation of functions

Recall that a function of bounded variation, also known as a BV-function, is a real-valued function whose total variation is bounded (finite).

Being more formal, the total variation of a real-valued function f, defined on an interval [a,b]R is the quantity:
Vba(f)=supPPnP1i=0|f(xi+1)f(xi)| where the supremum is taken over the set P of all partitions of the interval considered. Continue reading A continuous function which is not of bounded variation