Suppose thatL R2 → R is an continous function and that C is țhe grapco a...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
Analysis on Metric Spaces Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u 70, but f is not differentiable at (1,1). Prove that your choice of f has these properties (8 points).
Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u 70, but f is not differentiable at (1,1). Prove that your choice of , has these properties
Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u 70, but f is not differentiable at (1,1). Prove that your choice of , has these properties
Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u # 0, but f is not differentiable at (1,1). Prove that your choice of f has these properties
Please write clearly Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for sotne a >D and C>0, we have for any z.y e 0, 1 that Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for sotne a >D and C>0, we have for any z.y e 0, 1 that
2. Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u # 0, but f is not differentiable at (1,1). Prove that your choice of f has these properties (8 points).
l. Assume that j : R-→ R-s C and satisfies what are known as the Cauchy-Riemann equations: (c) Let r-(r1, 2) and (s1, s2) be vectors in IR2 and suppose that (ri, 2)f(s1, 82) and Df(81,82)メ0. Show that f-1 satisfies the Cauchy-Riemann equations when evaluated at r. (Hint: Might I make a notational suggestion: Leta(s) = sim) = % (n, s) and b(s) 쓺(81, 82) =-警( )) 81,82 (d) For this last bit, drop the assumption that f satisfies the...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Support function. The support function of a set C C R is defined as We allow Sc(y) to take on the value too.) Suppose that C and D are closed convex sets in R". Show that C D if and only if their support functions are equa Support function. The support function of a set C C R is defined as We allow Sc(y) to take on the value too.) Suppose that C and D are closed convex sets in...