Burkill began his secondary education at Richmond county school which he entered in 1911. Three years later he entered St Paul's School, London, where he was an outstanding scholar of classical Greek and Latin as well as showing considerable talent for mathematics. After winning a scholarship to Trinity College, Cambridge in 1917 he spend a short time in the Royal Engineers as World War I had not ended. However he was soon commissioned but then demobbed after the war ended. He continued with his education entering Trinity College, Cambridge in January 1919. After taking his first degree in 1921 he remained at Cambridge as a research student being elected to a fellowship in the following year after submitting a dissertation on surface areas. He then won a Smith's prize in 1923 for Functions of intervals and the problem of area.
In 1924 Burkill accepted the chair of pure mathematics at Liverpool and two years later Besicovitch joined him on the staff. He married Margareta Braun on 1 August 1928 and, at her insistence, he was known to all as Charles from that time on. Greta Braun (as she was called) had been born in Germany, having a German father and a Russian mother. Greta and Charles had three children, a son and two daughters. In 1929 Burkill returned to Cambridge taking up a a university lectureship and a lectureship at Peterhouse where he was also elected to a fellowship. He remained in this post for the rest of his life.
After Hitler came to power in Germany in 1933 there were many refugees who came to England and the Burkills did amazing work supporting refugee children :-
From 1933 onwards [Greta Burkill] helped to bring out of Germany and settle in England many hundreds of refugee children, and the Burkills themselves took into their family and assumed responsibility for the education of a German and an Austrian boy, who both went on to achieve positions in university departments of mathematics ... Several other children became for a time in effect members of the Burkill family while being helped to build new lives.In fact the years of World War II saw Burkill with tasks which gave him no time to continue with his mathematical research. He remained at Cambridge while many others took up war service, but he now had to take on administrative duties to cover for his absent colleagues. He also joined the university training corps and had reached the rank of major in the Royal Engineers section by the time the war ended. He resumed his research in mathematics winning the Adams prize in 1948 for an essay on integrals and trigonometric series. His research achievements were recognised in 1953 when he was elected to a fellowship of the Royal Society.
Burkill is equally well known for his research in analysis and the excellent teaching books which he wrote. He research is introduced in  by saying:-
Burkill's work is all in the theory of functions of a real variable with its main emphasis on theories of differentiation and integration. This was a particularly active area of research in the early decades of this century after the pioneering work of Lebesgue, Borel and their contemporaries in establishing the concepts of measure and the Lebesgue integral associated with it.Burkill introduced what is now called the 'Burkill integral' and applied it to extend W H Young's work on the definition of the area of a curved surface. He introduced the notion of approximate differentiation extending and simplifying work of Besicovitch.
Among his books are The Lebesgue integral (1951), A first course in mathematical analysis (1962) and A second course in mathematical analysis (1970). These are described in  as follows:-
These all display, as we might expect, not only his mastery of the field but a lucidity and elegance that encourages his readers to appreciate the profound aesthetic quality of good mathematics.Burkill himself sets out is aims in writing The Lebesgue integral in the introduction to the text:-
My aim is to give an account of the theory of integration due to Lebesgue in a form which may appeal to those who have no wish to plumb the depths of the theory of real functions. There is no novelty of treatment in this tract; the presentation is essentially that of Lebesgue himself. The groundwork in analysis and calculus with which the reader is assumed to be acquainted is, roughly, what is in Hardy's "A course of pure mathematics "(1908).A second course in mathematical analysis is described by T M Apostol in a review as a:-
... well-written text is designed as an introductory course in real and complex analysis for students familiar with elementary calculus and linear algebra.The book covers: sets and functions, metric spaces, continuous functions on metric spaces, real and complex limits and series, uniform convergence, Riemann-Stieltjes integration, multivariable differential and integral calculus, Fourier series, Cauchy's theorem, Laurent expansions, residue calculus, infinite products, the factor theorem of Weierstrass, asymptotic expansions, and applications to special functions in particular the gamma function.
In 1961 Cambridge promoted Burkill to be Reader in Mathematical Analysis. He retired in 1967 and a year later Sir Herbert Butterfield retired as Master of Peterhouse. Burkill was appointed to succeed Butterfield as Master and he served from 1968 until 1973 :-
... it was as keeper of the college and its traditions that Burkill saw his role. Coming to office at a time of widespread student unrest, he took a far-sighted view of its eventual course, and devised effective machinery for consultation and dialogue with junior members without saddling the college with a statutory commitment to full participation in its government more easily incurred than shed. His relationship with the fellows was also based on consultation, and in particular on cultivating a spirit of trust and co-operation between the master and the tutors ...After retiring as Master, he served as editor of the Mathematical Proceedings of the Cambridge Philosophical Society. However he continued his interest in integration theories and published a short paper Fourier-Stieltjes integrals in 1973. It examines problems in Fourier analysis that led to the development of the theory of generalised functions.
In  Burkill is described as having a:-
... fastidious concern for accuracy and economy in the use of words. In respect of the spoken word, this economy became something of a legend. Taciturn is not a sufficiently friendly word to describe his conversational style, because it contained no hint of malice or lack of concern but only an unerring judgement about what was important, and the clearest way of saying it. What is even more important is that his distaste for excessive display of feeling concealed, at first, a truly generous and hospitable nature.His character is described in  as follows:-
Brevity, precision, and conciseness were as characteristic of Burkill's conversation as of his written style, and his silences could be eloquent. He was an accomplished mimic. As a boy he had been given to practical joking, and though in later life his sense of humour was well under control it was never far from the surface. Fellows of Peterhouse were familiar with (and were indeed known to imitate) the slight sideways vibration of the body which betokened amusement and often precluded a mildly ironical or deflationary quip.Greta Burkill died in 1984 and Burkill did a large amount of work in putting her papers on her refugee work into order so that they could be deposited in the Cambridge University Library. We mentioned above that the Burkills took over the responsibility of educating two refugee children who went on to become professional mathematicians. One of these was Harry Burkill who was on the staff at Sheffield University. After his health failed, Burkill lived in a Sheffield nursing home. He died there of bronchopneumonia although he had suffered from Alzheimer's disease for some time. After he was cremated his ashes were scattered at Hutcliffe Wood crematorium in Sheffield.
Article by: J J O'Connor and E F Robertson