Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval. Prove that f(x) and xf(x) are linearly independent on II.

Let us define a number a to be a fixed point of a function f if a = f(a). (For example, if f (x) = -x², then x = -1 is a fixed point because f(–1) = -1.) Prove that if f'(x) # 1 for all numbers x, then f has at most one fixed point.

determine the continuity of the function at a.

Show that the function f : R → R given by f(x) = x/(1+|x|) is increasing.