Consider the vector space R^2 with the norm ║(x,y)║ = │x │+│y │

Consider the vector space R^2 with the norm ║(x,y)║ = │x │+│y │ Show that the set U = { u element of R^2 : 0< ║u║ < 1} is an open set in this normed vector space.

Suppose that A = R^2 with {(0,0)} removed and that f :A→ R is a uniform continuous mapping on A.

Suppose that A = R^2 with {(0,0)} removed and that f :A→ R is a uniform continuous mapping on A. a)Prove that there exists L an element of R so that lim f (x,y) = L [(x,y) → (0,0), (x,y) element of A]. b)Using L from part (a) prove that F(x,y) = { f(x,y) when (x,y) ≠ (0,0) and L when (x,y) = (0,0)} edit: for some reason it will not post the backslash sign . . . i'm sorry. A is R^2 with a hole at the origin, or {(0,0)} removed.

Is there any f: R->R such that f'(x) = g'(x)?

See Attachment.

If f'(a) exists then

See Attachment.

Taylor's Theorem

See Attachment.