Show that `f (x) = |x|` is continuous at x= 0 but f'(0) does not exist

Prove that f (x) =|x| is continuous at x =0

Show that f(x) = x-[x] is discontinuous for integral values of x

if f(x+y)=f(x)+ f(y) for all x and y and if f(x) is continuous at x=0, show that f(x) is continuous everywhere.

if f(xy) = f(x) f(y) for all x and y and if f(x) is continuous at x = 1, then show that it is continuous for all x except 0

Show that f (x) = f(x) = {(x,sinfrac(1)(x)),(0,x=0):} is continuous at x = 0

If the function f(x)={{:((sin x)/(x) + cos x", if " x ne 0 ), (k", if " x =0 ):} is continuous at x=0, then find the value of k.

Show that f (x)= x^2 is strictly decreasing in (-oo,0)