Question

नि म्न पद्यांश पढ़कर पछू गए प्रश्नों केउत्तर लि खि ए l 5m
सुनकर बोली और -और
कठपतु लि यां कि हां,
बहुत दि न हुए
हमअपनेमन के छं द छूए
मगर ,
पहली कठपतु ली सोचनेलगी
ए कैसी इच्छा
मेरेमन मेंजगी?
1. यह पद कि स पाठ सेलि या गया है? कवि कौन हैं?
2. पहलेकठपतु ली की बात दूसरी कठपतु लि यां को अच्छा क्यों लगी ?
3. पहली कठपतु ली अब क्या सोचनेलगी ?

Answer 1

Answer:

But let’s find some less trivial examples of functions  f:R→R  that satisfy this functional equation.

Now, let’s choose any number  x∈R , and try to fix  f(x)=α≠1 . What does it tell us?

etc. And, as long as all those numbers are different, and different than 1, we’ll find no contradiction. We can easily make that happen by selecting a transcendental  α .

We can go one step further, and define  f(2kx)=α , for all  k∈Z . Since  α  is transcendental, we can see that all numbers of the form  2aαbx  are different, and all their images would have thus been fixed.

Now, if we define  Sx={y∈R|∃a,b∈Z,y=2aαbx} , the  Sx  are classes of equivalence that partition  R . Now, just pick one element of each of the classes (using the axiom of choice or otherwise), and define that  f(2kx)=α  for all of those  x , and all  k∈Z . That defines  f  uniquely and consistently in all  R .