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

Prove that f (x) = [X] is discontinuous for integral values of x.

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

if f (x) = x^3 , show that f''(x) exist for all real values of x and find it.

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.

If f(x) = k, a constant, find the value of Deltaf(x) .

Show that the function f, defined by f(x) ={(2x, x 2):} is discontinuous at x = 2.

Show that f(x)=e^(1//x) is a strictly decreasing function for all x in R .

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

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