Recent content by kbgregory

Ok, I just found out from my professor that there is a typo in the problem, and that f is continuous on [0,1], so I think I can take it from here.

Homework Statement Suppose that f is continuous on (0,1) and that int[0,x] f = int[x,1] f for all x in [0,1]. Prove that f(x)=0 for all x in [0,1]. Homework Equations We know that since f is continuous on (0,1), F(x) = int[0,x] f and F'(x) = f(x) for x in (0,1). The Attempt...

Homework Statement Show that the function f(x) = { x/2 if x is rational { x if x irrational is not differentiable at 0 Homework Equations If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e...

Homework Statement If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h Homework Equations We know that f'(a) = lim[h->0] (f(a+h)-f(a))/h The Attempt at a Solution I have done the following, and I am not sure if it is correct, though the result makes sense...

Homework Statement Suppose f is differentiable on an open interval I and let x* \in I. Show that there exists a sequence {x_n}\subset I such that lim[n->inf]{x_n}=x* and lim[n->inf]{f'(x_n)}=f'(x*). Homework Equations We know that a function g is continuous iff for any sequence...

Wow, thanks a lot! Got it.

Homework Statement Suppose f is a continuous function on [0,2] with f(0) = f(2). Show that there is an x in [0,1] where f(x) = f(x+1). Homework Equations By the Intermediate Value Theorem, we know that any values between sup{f(x)} and inf {f(x)} over x in [0,2] will be repeated...