of Continuous Functions. 
525 
positive. This proves that an upper semi-continuous function 
which is bounded and never negative, can be expressed as the 
limit of a monotone decreasing sequence of continuous bounded 
positive functions. 
Case 2. Let f(x) be lower semi-continuous, bounded, and 
never negative. 
We have in the preceding discussion only to use the lower 
limit instead of the upper limit, and so express f(x) as the limit of 
a monotone increasing sequence of continuous bounded functions 
which are never negative. It is to be remarked that we cannot 
now ensure the generating function being always -f, since the 
addition of a decreasing sequence of small quantities such as was 
used before might destroy the monotony of the sequence. 
Case 3. Next let f{x ) be any upper semi-continuous function. 
At every point where f is positive put 
u (x) =f(x) + A, v (x) = — A, 
and elsewhere 
u(x) = A, v(x) = f(x) — A. 
Thus at every point 
u ( x ) 4- v (x) = f (&■). 
It is obvious that u and v are both upper semi-continuous 
functions, one of which is positive and ^ A and the other negative 
and ^ — A everywhere. 
Now u is always ^ A, and v ^ — A. Thus 
- is lower semi-continuous and lies between 0 and \ , 
u A 
- is lower semi-continuous and lies between 0 and r , 
v A 
which makes 
— ^ upper semi-continuous, and lying between 0 and . 
Thus, by case (2), 
— Lt u n , 
a w = co 
where u n ' is a continuous bounded function which is never 
negative, and the sequence is monotone increasing ; whence, 
u= Lt —7 = Lt u n , say, 
n = co Un m = 00 
where u n is a continuous (but in general unbounded ) positive 
function, and the sequence is monotone decreasing. 
