prove thsh f(x,y) then n exists 3, 、 2 2. prove thsh f(x,y) then n exists 3, 、 2 2.
Problem 1. (2 credits) Let f: X +Y. Prove that f is injective if and only if there exists a function g: Y → X such that go f = ldx.
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
(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...
1. Suppose that f : NR. If lim f(n+1) f(n) = L n-oo prove that lm0 S (n)/n exists and equals L
1. Suppose that f : NR. If lim f(n+1) f(n) = L n-oo prove that lm0 S (n)/n exists and equals L
If n is greater than m, prove that there exists no injective map f: Sn --> Sm; where Sn = {1,....,n}. You must prove this by induction.
Prove (X - Y) nZ = (X n 2) - Y
3. Find lim f(,y) if it exists, and determine if f is continuous at (0,0. (x,y)--(0,0) (a) f(1,y) = (b) f(x,y) = { 0 1-y if(x, y) + (0,0) if(x,y) = (0,0) 4. Find y (a) 3.c- 5xy + tan xy = 0. (b) In y + sin(x - y) = 1.
2. Prove that if X, Y have a joint density, then for any Be B, f(y, x) JB f(x)
2. Prove that if X, Y have a joint density, then for any Be B, f(y, x) JB f(x)
Given f(x,y) = 2 ; 0 <X<y< 1 a. Prove that f(x,y) is a joint pdf b. Find the correlation coefficient of X and Y
= (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].