" Show that "f(x,y)(xy)/(x^(2)+y^(2)),x=y!=0" is discontinuous at origin when "f(0,0)=0

Show that the funcation f(x)={{:(3x-2",", " when ", x le0),(x+1",", " when ", x gt 0):} is discontinuous at x =0

f(x)={{:(3-x",","when",x le0, ),( x^(2)",", "when",xgt0):} is discontinuous at x=0

Consider f(x) is a real valued function f(x) such that f(x+y)=f(x)+f(y)+2xy, such that f(x) is differentiable for all xinR and (df)/(dx)(0)=0,then answer the following

Let g(x,y) = (x^(2)y)/(x^(4) + y^(2)) for (x,y) ne (0,0) and f(0,0)=0. (i) Show that lim_((x,y) to (0,0)) g(x,y)=0 along every line y=mx, m in R . (ii) Show that lim_((x,y) to (0,0)) g(x,y) = k/(1+k^(2)) , along every parabola y=kx^(2), k in R{0} .

A function f:RtoR is such that f(x+y)=f(x).f(y) for all x.y inR and f(x)ne0 for all x inR . If f'(0)=2 then f'(x) is equal to

Let (f(x+y)-f(x))/(2)=(f(y)-1)/(2)+xy , for all x,yinR,f(x) is differentiable and f'(0)=1. Let g(x) be a derivable function at x=0 and follows the function rule g((x+y)/(k))=(g(x)+g(y))/(k),kinR,kne0,2andg'(0)-lambdag'(0)ne0. If the graphs of y=f(x) and y=g(x) intersect in coincident points then lambda can take values

Let (f(x+y)-f(x))/(2)=(f(y)-1)/(2)+xy , for all x,yinR,f(x) is differentiable and f'(0)=1. Let g(x) be a derivable function at x=0 and follows the function rule g((x+y)/(k))=(g(x)+g(y))/(k),kinR,kne0,2andg'(0)-lambdag'(0)ne0. If the graphs of y=f(x) and y=g(x) intersect in coincident points then lambda can take values