Leg `f(x+y)=f(x)+f(y)fora l lx , y in R ,` If `f(x)i scon t inuou sa tx=0,s howt h a tf(x)` is continuous at all `xdot`

Leg f(x+y)=f(x)+f(y)fora l lx , y in R , If f(x) is continue at x=0,s howt h a tf(x) is continuous at all xdot

Let f(x+y)=f(x)+f(y) for all xa n dydot If the function f(x) is continuous at x=0, show that f(x) is continuous for all xdot

(i) if f ( (x + 2y)/3)]= (f(x) +2f(y))/3 , " x, y in R and f (0) exists and is finite ,show that f(x) is continuous on the entire number line. (ii) Let f : R to R satisfying f(x) -f (y) | le | x-y|^(3) , AA x, y in R , The prove that f(x) is a constant function.

Let f:R to R be a function satisfying f(x+y)=f(x)+f(y)"for all "x,y in R "If "f(x)=x^(3)g(x)"for all "x,yin R , where g(x) is continuous, then f'(x) is equal to

Let f(x+y)+f(x-y)=2f(x)f(y) AA x,y in R and f(0)=k , then

If f(x+y)=f(x) xx f(y) for all x,y in R and f(5)=2, f'(0)=3, then f'(5)=

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

Let f(x+y)=f(x)+f(y) and f(x)=x^2g(x)AA x,y in R where g(x) is continuous then f'(x) is

Let f:R to R be a function given by f(x+y)=f(x)f(y)"for all "x,yin R If f(x) ne 0 for all x in R and f'(0) exists, then f'(x) equals