Prove that f(x) = sin x and f(x) = cos x are continuous.

Prove that the identity function f(x) = x is continous

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

Prove that function f given by f (x) = log ( cos x ) is strictly decreasing on (0, (pi)/(2)) and strictly increasing on ((pi)/(2) , pi).

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

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

Prove that f : R rarr R, f(x)=4x-5 is invertible and find f^-1(x) .

Prove that f (x) = |x-1|,x in R is not differnentiable at x=1

Prove that int_(0)^(a) f(x) dx = int_(0)^(a) f(a-x) dx . Hence, evaluate int_(0)^(pi//2) (sin x - cos x)/(1+ sin x cos x) dx.

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.