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Every continuous function is differentiable.
True or false with full explanation
-2 is the limit point of the interval ]-3,2[.
True or false with full explanation
Show that series Σ x/(1+n^2.x^2) is uniformly convergent in [k,1] where k>1 but not uniformly convergent in [0,1]
Let f : M → Y be a homeomorphism and O an open subset of M. Explain
concisely in no more than two lines of text why f(O) is an open set.
Let f : M → Y be a homeomorphism and O an open subset of M. Explain
concisely in no more than two lines of text why f(O) is an open set
Let k ≥ 0 and f : M → M a k-Lipschitz function. Let ε > 0. Give the largest
number φ > 0, if any, such that ∀x, y ∈ M, d(x, y) < φ implies d(f(x), d(y)) < ε.
Show the series Σ x/(1+n^2.x^2 is uniformly convergenr in [ᾰ,1], for any ᾰ>1.
Show that the series Σ x/(1+n^2.x^2) is uniformly convergent using Weierstrass m test
Suppose that the sequence (sn) converges to s and sn ≤ A for every n. Show that s≤A
Let f: [0,1] to R ne a function defined by
f(x)= 1-x^2
Let P1= { 0,1/2,2/3,1}
P2= { 0,1/4,1/2,3/4,1}
be two partition of the interval [0,1]. Calculate L(P2,f) and U(,P1,f)
