(a) Is there any continuous function f:R->R such that f(f(x))=e^(-x) for every real x ?
(b) Is there any continuous function f:R->R such that f(f(x))=e^x for every real x ?

It took me less than half an hour to answer the first and more than a year to answer the second...

Phantazomai oti paragwgises me kanona alusidas kai prosap8hses na epilyseis th diaforikh pou proekypse....Etsi;

H f den einai anagaia paragogisimi! Alla akoma kai an theso auto san upothesi h diaforiki den einai epilisimi...

Swsto.Opote pas thn tyhi????Einai dyskolo alla einai spazokefalia....Oxi ma8hmatika aparaithta.Ektos an exeis vrei formula pou na lynei kapoies synarthseis me fof=g(x).Tote pragmatika ekanes kati kalo....

Gia kapoies g(x) vrika. Stin proti periptosi den uparxei tetoia f giati:
1)H f einai 1-1 (eukolo...)
2)Afou i f einai kai sinexis tote tha einai monotoni.(ligo pio diskolo...)
3)Sinthesi monotonon einai panta auxousa(eukolo...). Alla i e^(-x) einai fthinousa!

Ok.Swsth skepsh!Vrhkes kai lush sth fof=exp(x) h apedeikses kai se aythn thn periptwsh oti den ypaxei tetoia f;

Gia peripou enan xrono prospathousa na deixso oti den uparxei alla telika uparxei :Esto a <0. Orizo I1=(-Inf,a), I2=(a,0), I3=(0,e^a),...,In=e^I(n-2),... kai
f1:I1->I2 , f1(x) = a-ae^(x-a)
f2:I2->I3 , f2(x) = e^(invf1(x)) inv=antistrofi
f3:I3->I4 , f3(x) = e^(invf2(x))
...............
fn:In->I(n+1) , fn(x) = e^(invf(n-1)(x))
..............
H f einai kladiki me kladous tis fi. Tin apodixei den xoraei na tin grapso...

Mravo ypomoni kai skepsi!!!Egw na sou pw thn alh8eia 8a ta eixa parathsei apo thn prwth mera!!!!