of Continuous Functions. 
521 
For let P be any point, and Q a point on the left of P. Then, 
since f n O) is a continuous function, there is a point x n of the 
closed interval (P, Q) at which f n (x) assumes its least value in 
the interval (P, Q), say 
fn On) — Ln, q 
>f+i On) fn + 2 On) ^ 
>fOn) 
^ lower bound of f(x) in the interval (P, Q). 
Allowing n to increase without limit, we see that the lowest 
possible limit Lq of L n q is also ^ the lower bound of f(x) in 
Of Q). 
Now moving the point Q up to P as limit, Lq has, by 
the definition of the left-hand chasm function, the limit %i(P), 
while the lower bound of f(x) in (P, Q ) has, by the definition 
of the left-hand associated lower limiting function, the limit 
^ l(P ), thus 
(-P). 
Now it being known that in all cases, 
XL (P) < (P) ^ $L (P) ^ 7T£ (P), 
we get the first of the relations (3), viz. 
XL (P) = ^ L (P) < <t>L ( P ) < 7T& 00> 
(f) L being the associated upper limiting function and tt l the peak 
function on the left. 
To obtain the second of the relations (3), we have in like 
manner a point xf of the interval (P, Q) at which f n {x) assumes 
its greatest value in (P, Q), say 
fn Of) = M n , Q 
^ fn — l On ) ^ M-n—i, Q- 
Hence d/ ?l) q q J\T n +- 2 , q ^ ^ d / q ^ ir^ 0 ), 
and therefore, fnOn) ^ 7r^(P). 
Now let n be fixed and Q move up to P as limit, then the 
point x n \ if not already the point P itself, moves up to P as limit, 
so that, f n O) being continuous, 
fn ( P ) > TTL (P)> 
Now let n increase without limit, and we get 
f(P)>TT L (P), 
whence, it being known that in all cases, 
/(P) ^ 7 t l (P), 
