WebLet $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$. I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the ... WebDec 27, 2015 · Sorted by: 10. Set f ( x) = g ( x) + x 2 2 then plugging in gives. 1 g ( x + y) = g ( x) + g ( y). This is Cauchy's functional equation. And under certain regularity …

Cauchy

WebAssume that (1) f (x+y)+ f (xy) = f (x)+f (y)+f (x)f (y) for all x,y ∈ R. As others have noticed, an obvious solution is f ≡ 0, so we assume from now on that f is ... If a is a web page, let V (a) be the set of people who have visited a. Then a,b ∈ R if and only if V (a) ⊆ V (b). WebSolution Verified by Toppr Correct option is C) We have, f(xy)=f(x)+f(y)⇒(1) Put x=y=1 ⇒f(1)=0 Now f(x)= h→0lim hf(x+h)−f(x)= h→0lim hf[x(1+h/x)])−f(x) = h→0lim hf(x)+f(1+h/x)−f(x)= h→0lim h/xf(1+h/x)−f(1)⋅ x1= xf(1) Now integrating we get, f(x)=f(1)logx+c Since f(1)=0⇒c=0 Thus f(x)=f(1)logx Also f(2)=1⇒1=f(1)log2⇒f(1)= log21 mbira music is from

3. Let F (x,y)= xy,x−2y and let C be the piece of Chegg.com

Web{\displaystyle f(x+y)=f(x)+f(y).\ A function f{\displaystyle f}that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebrathat there is a single family of solutions, namely f:x↦cx{\displaystyle f:x\mapsto cx}for any rational constant c.{\displaystyle c.} A simple argument, involving only elementary algebra, demonstrates that the set of additive maps , where are vector spaces over an extension field of , is identical to the set of -linear maps from to . Theorem: Let be an additive function. Then is -linear. Proof: We want to prove that any solution to Cauchy’s functional equation, , satisfies for any and . Let . mbi scrapbook how to refill pages