Advanced book on Mathematics Olympiad

128 3 Real Analysis

381.Does

lim

x→π/ 2

(sinx)

cos^1 x

exist?

382.For two positive integersmandn, compute

lim

x→ 0

√mcosx−√ncosx

x^2

383.Does there exist a nonconstant functionf :( 1 ,∞)→Rsatisfying the relation

f(x)=f(x

(^2) + 1
2 )for allx>1 and such that limx→∞f(x)exists?
384.Letf:( 0 ,∞)→( 0 ,∞)be an increasing function with limt→∞f(f(t)^2 t)=1. Prove
that limt→∞f (mt)f(t) =1 for anym>0.
385.Letf(x)=
∑n
k= 1 aksinkx, witha^1 ,a^2 ,...,an∈R,n≥1. Prove that iff(x)≤
|sinx|for allx∈R, then
∣
∣∣
∣∣
∑n
k= 1
kak

∣

∣∣

∣∣≤^1.

3.2.2 Continuous Functions.....................................

A functionfis continuous atx 0 if it has limit atx 0 and this limit is equal tof(x 0 ).A
function that is continuous at every point of its domain is simply called continuous.

Example.Find all continuous functionsf:R→Rsatisfyingf( 0 )=1 and

f( 2 x)−f(x)=x, for allx∈R.

Solution.Write the functional equation as

f(x)−f

(x

2

)

=

x

2

;

then iterate

f

(x

2

)

−f

(x

4

)

=

x

4

,

f

(x

4

)

−f

(x

8

)

=

x

8

,

···

f

( x

2 n−^1

)

−f

(x

2 n

)

=

x

2 n

.