TRANSACTIONS OF SECTION A. 4A5 
kind has been devised in which there is nothing electrical, the thermometric 
receiver, instead of a thermo-element, being a bimetallic spiral, which deflects 
a pointer over a scale attached to the instrument. 
DEPARTMENT OF MATHEMATICS. 
The following Papers were read :— 
1. An Account of Modern Work on the Calculus of Variations. 
By Professor A. R. Forsyrn, 7.2.8. 
2. Some New Results in the Theory of Functions of a Real Variable. 
By Dr. W. H. Youne, F.#.S. 
I. The Distinction of Right and Left at a Point of Discontinuity, 
1. Definition of the Associated Functions.—Let P be any internal point of a 
segment -in which f is defined. Take an interval with P as right-hand end- 
point; then f() has, for points internal to this interval, an upper limit; as the 
interval decreases this upper limit cannot increase, and therefore has a limit 
gx(P). This function is called the upper left-hand limiting function of f. 
Similarly we define the upper right-hand limiting function @z, and the lower 
left-hand and right-hand limiting functions Wy and Wr. 
Further, at each point we select that one of the two upper functions which is 
not less than the other; the function so defined is called the (modified) upper 
limiting function ¢ 
The word modified is introduced to mark the distinction between this function 
and what I have elsewhere called the upper limiting function, and which I now 
call Baire’s upper limiting function, and denote by dg. This is that one of f and 
¢ which is not less than the other. 
Similarly, changing less to greater and upper to lower, we have the lower 
limiting function yp. 
2, If f is bounded (borné), so are the limiting functions. If not they will have 
proper infinite values, even when / is always finite; e.g., f=0 at irrational points, 
f=q at any rational point ”. 
It is easily proved that the points where one of the upper limiting functions 
= + © form a closed set, and where it = — oo an inner limiting set. 
3. Theorem 1,.'—Any limit approached by any upper limiting function (modified 
or not) as x approaches a point P as limit on the left < pr (P), and as v approaches 
P as limit on the right < $1 (P). 
Cor. 1.—@x is upper semi-continuous on the le{t and ¢p on the right, while ¢, 
like dz, is an upper semi-continuous function, and, as such, at most pointwise 
discontinuous, 
Cor. 2.—At any point where ¢.=¢r=¢, both pr and gp are upper semi- 
continuous (on both sides). (N.B.—The converse is not true. ) 
4, Theorem 2.—At every point of continuity of ¢, ¢,=¢r=¢, and both 
gr and dp are continuous. 
Cor..—All the associated upper limiting functions are at most pointwise dis- 
continuous. 
(N.B.—Either of the unsymmetrical upper functions may have other points of 
continuity. Example, /=0, except at a sequence on the right of its limiting 
point, where f= 1.) 
Theorem 3,—The points of continuity of dp are among those of ¢, so that at 
1 In the statements of theorems we usually confine ourselves to the d’s. The 
corresponding statements for the y’s are obvious. 
