Let f :R →R be a function defined by f (x + y) = f (x) f ( y) for all x, y∈R. If f (x) is continuous at
x = 0, show that f (x) is continuous for all x∈R
62a hint...
1) use f(0+y)=f(0).f(y)
2) prove lim h->0 (f(x+h)-f(x)) =0
Using 1, 2 is fairly simple.
If u dont get it nikhi.. do tell me I will give the complete solution.
1thanks a score...i have got the idea...
but what should be our approach while solving these kind of problems..
or problems based on functional relation...is there some general pattern regarding these kind of questions.???
got hooked on these question...
let f{x} be a continuous function in [-1,1] and satisfies f{2x2-1}=2x f{x} for x belongs to [-1,1].prove that f{x} is identically zero for all x belong to [-1,1].
62hmm...
see it is a bit tricky...
With these functions, generally u will (may be) able to find the values of f(0) or f(1) .. This is a mojor thing.. by some substitution!)
Then we have to generally work based on the definition..
Example if it is continuous. then we have to use the first principle of continuity... other wise differentiability definition (the most basic one using limits)
Many other problems are simply tricks.. substituting x by 1/x or some other thing...
So there is no hard and fast way... It is all about the trick striking ur brain!
62For your problem,
let f{x} be a continuous function in [-1,1] and satisfies f{2x2-1}=2x f{x} for x belongs to [-1,1].prove that f{x} is identically zero for all x belong to [-1,1].
x=1 substitution gives f(1)=0
so we have somthing to work with!!!
What could be the next ideas?
if we could prove the derivative to be zero?
I am short of more ideas immediately! but will try to solve this one today and post the solution...