258 
Proceedings of the Royal Society of Edinburgh. [Sess. 
monotone sequence of semi-continuous functions, viz. of a monotone 
decreasing (increasing) sequence of upper (lower) semi-continuous functions. 
These are particular cases of functions which are pointwise discontinuous 
with respect to every perfect set, and have, moreover, the advantage that 
they may be generated as the limits of monotone sequences of continuous 
functions. To prove that a function is pointwise discontinuous with 
respect to every perfect set, it follows that it is sufficient to show that it is 
the limit of (1) a monotone decreasing sequence of lower semi-continuous 
functions, and (2) of a monotone increasing sequence of upper semi- 
continuous functions. 
It is interesting to remark that what I call the upper function of a 
sequence of continuous functions (that is, the function which at each point 
has the value of the highest possible limit approached by the values of the 
continuous functions there) satisfies the condition (1), while the lower 
function satisfies (2). This may be proved in the manner indicated in 
Hobsons Functions of a Real Variable, p. 552, line 22 seq., where it is 
shown that the upper derivate, which is a particular case of an upper 
function, is the limit of a monotone decreasing sequence of functions 
w 1 ,w 2 , . . . . each of which is the limit of a monotone increasing sequence 
of continuous functions v 1 , v 2 , . . . . 
It follows that an upper (lower) function of a sequence of continuous * 
functions, and in particular an upper lower derivate, is upper (lower) 
semi-continuous except at the points of a set of the first category (viz. the 
points at which one at least of the semi-continuous functions of the mono- 
tone sequence is discontinuous). This is true, moreover, not only with 
respect to the continuum, but with respect to every perfect set. 
This, again, gives us a new proof of Baire’s theorem, that (in the case 
when the upper and lower functions coincide) the limit of a sequence of 
continuous functions is pointwise discontinuous with respect to every 
perfect set. The converse, proved also by Baire, viz. that any function 
which is pointwise discontinuous with respect to every perfect set is the 
limit of a continuous function, shows that the conditions above given are 
not only sufficient but necessary for the function in question to have the 
required property. 
* It may be added that this result can be still further generalised. The general result 
is that the upper (lower) function of a sequence of lower (upper) semi- continuous functions 
is upper (lower) semi-continuous except at the points of a set of the first category. Further, 
this is true with respect to the continuum or any perfect set. 
{Issued separately April 10, 1908.) 
