Define g : [0, 1] → R byg(x) =sin(x2)x if 0 < x ≤ 1,0 if x = 0.13. Prove that g ∈ R[0, 1]

Let f : [0, π] → R be defined byf (x) =(0 if x = 0,x sin 1x − 1x cos 1x if x̸ = 0.Is f continuous?

Define g : R → R byg(x) =(x2 cos(1/x) if x̸ = 0,0 if x = 0.Show that g is differentiable at x for every x ∈ R

Define f, g : R → R by f (x) = (x − 2)2 andg(x) =1 if x > 0,0 if x = 0,−1 if x < 0.Calculatelimx→2 g(f (x)) and g limx→2 f (x)

Show that the following functions are differentiable at x = 0.(a) |x| sin x (b) x2/3 sin x (c) 3√x(1−cos x) (d) h(x) ={x2 sin(1/x), x 6 = 00, x = 0.Page 3

For each n ∈ N, define fn : [0, 1] → R byfn(x) =(xn cos(1/x) if x > 0,0 if x = 0.11. Prove that fn ∈ R[0, 1] for all n ∈ N