Suppose that (a-r, a) C E or (a, a + r) C E, f : E → R, L E R, and (1) Prove that there exist num...
4. Suppose f : D → R is a function and a ∈ R, and that for some β > 0, D contains (a-β, a + β)-{a} = (a-β, a) U (a, a + β). Prove that limx→a f(x) = L if and only if for all ε > 0 there exists δ > 0 such that if 0 < lx-al < δ and x ∈ D, then If(x) - L| < εDefinition: Suppose f : D → R is a function, a...
(a) Suppose that lim x→c f(x) = L > 0. Prove that there exists a δ > 0 such that if 0 < |x − c| < δ, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then there exists an " > 0 such that f(x) ≥ " for all x ∈ [a, b]. = (a) Suppose...
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If f is differentiable on R", prove that af Экп af axi (Xi , . . . , xn) ER". for all x Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If...
need help with all a, b, c 2. 15 Marks (a) Suppose that f : R" R is convex but not necessarily smooth. Prove that h-af is a (b) Suppose that f : R -R is convex and smooth. Also assume that f(x) > 0 for all z (c) Show that the set S = {(x,y) : y > 0} is convex and that the function f(x,y)-x2/v is convex function if a-0. Show with a simple example that this is...
1. Suppose m,b,c E R. Prove: f(1) = mx + b is continuous at c. 2. Prove: f(x) = x3 is continuous at 5.