The problem dealt with in this paper has been raised by P Turàn. To every point of the interval 0 ≤ x ≤ 1 we adjoin a finite number of points of the same interval. This means that we define a function y = φ(x) for 0 ≤ x ≤ 1 where 0 ≤ y ≤ 1, φ(x) takes a finite number of values for every x, and the equation x = φ(x) is impossible. Two points x and y are called independent if neither of the two equations y = φ(x) and x = φ(y) holds.

To every point x we adjoin an interval containing x and having endpoints with rational coordinates, in such a manner that all values of φ(x) are situated outside of this interval. We assert that there is an interval which is adjoined to a set of points of the power of the continuum. This assertion follows immediately from the fact that intervals with rational endpoints are countably many, and from G König's theorem, which states that the power of the continuum cannot be the sum of countably many smaller powers. The points to which this interval belongs are evidently independent, for the adjoined values are all outside the interval.

We see that we did not make use of the fact that φ(x) has only a finite number of values for every x, but only of the fact that x is not a condensation point of the values of φ(x).

About Dezső Lázár I would like to say that when I moved to Cluj, he was there, working as a teacher in the Jewish Gimnázium. Later, he was called upon for forced labour service. He was made to detect mines, was wounded in the thigh [when a mine exploded], and left to bleed to death. While he was in the forced labour service, we [László Fejes Tóth and his wife] kept in close contact with his family. We often visited his wife and two small children. We learned about what happened to him from his wife. I don't remember exactly when this happened, because the years have become blurred in my memory. His wife was a refined, beautiful lady and the thought horrifies me to this day that [in 1944] she was dragged away in a box-car and after a lot of suffering they murdered her with her two small children in the gas chambers of Auschwitz.

The author proves the following theorem. To every closed convex curve there exists an inscribed n-sided polygon of area $t_{n}$ and a circumscribed n-sided polygon of area $T_{n}$ so that $(T_{n}- t_{n})/T_{n}≤ \sin^{2}(\pi/n)$.

The memory of Dezső Lázár did not go to waste thanks to the former memorial committee of the János Bolyai Mathematical Society. The committee, at the suggestion of Rózsa Péter, initiated in 1974 that secondary school mathematics clubs undertake to cherish the memory of a mathematician who died during the Second World War. The club leader of the Teleki Blanka High School in Székesfehérvár, Hugó Láng, chose Dezső Lázár from the list of names offered. During the research, Hugó Láng also wrote a letter to Pál Erdös, who wrote the following in his reply:

Wouldn't it be good to establish a small prize in memory of Lázár, which the best member of the class would receive every year or every second year. If you would consider this a good idea, I would be happy to cover the financial costs.

Thus, the Dezső Lázár Prize was established at the school, which is awarded to two students annually. The 1977 prizes were presented by Pál Erdös himself at the school.

There are few memorials in public places remembering the hundreds of thousands of victims of the Hungarian Holocaust and yet fewer, remembering individuals. There is a memorial plaque at the Rényi Institute of Mathematics in Budapest with two sets of names of martyrs. One is for "Our Greats" and the other for those who had just "Embarked on the Road of Creating." Dezső Lázár's name is carved into this second list.