True or False , I don't understand....?

Question

If a vector valued function r(t) = f(t) i + g(t) j + h(t) k is continuous for all real values of the scalar variable t, then r '(t) must exist for all real values of t.

Answer 1

No. continuity does not imply differentiabilty. If r(t) = ti + tj +|t| k, then r is continuous for every real t but r' does not exist at t=0, because the function h(t) = |t|, though continuous at t =0, is not diffrentiable at this point.

Answer 2

I believe the answer is false because continuity does not imply differentiability. This is because functions must be both continuous and smooth to be differentiated.

Consider f(x) = |x|. This function is continuous for all real values of x, but it is not differentiable at x = 0 because it is not a "smooth function."

Therefore, I imagine that this same principal applies for an even more complicated vector valued function.

Answer 3

I think this is false. If you look at a function like f(x) = abs(x), it is continuous but not differentiable at x=0. The point being, continuity doesn't guarantee differentiability. The fact that this is a vector values function shouldn't, to my knowledge, change that.