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`

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

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

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

(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(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

A function f(x) satisfies the following property: f(xdoty)=f(x)f(y)dot Show that the function f(x) is continuous for all values of x if it is continuous at x=1.

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

Statement-1: If |f(x)| le |x| for all x in R then |f(x)| is continuous at 0. Statement-2: If f(x) is continuous then |f(x)| is also continuous.

Let f(x) be a function such that f(x+y)=f(x)+f(y) and f(x)=sin x g(x)" fora ll "x,y in R . If g(x) is a continuous functions such that g(0)=k, then f'(x) is equal to

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)=