Prove the following Theorem: Theorem. f : X → R is continuous + for any open...
a) Prove the following theorem: Let f:(x,d)-(Y,p) be bijective and continuous. Then f is a topological mapping iff: VUCX: U open = f(U) open in Y. b) Þrove the following theorem: Let f :(X,,d) (X ,d) and f:(X2,d)) (X 3,d) be topological mappings, Then f of, (the composition of the two functions) is topological.
How to prove these two questions? 3. Let f : 10, 11 → [O, 1] be continuous. Then there exists x [0,1] such that f(x)-x. 4. A function f : R → R is continuous if and only if the pre-image of all open sets are open. Note: The pre-image of a set s is defined as f-1 (S)-{re R : f(x) є S). For example if f(x), then f ((0,1) (1,0)U (0, 1). =x- 1,0) U (0, 1
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
*4, Let U be an open subset of R" and f:U-R" a function whose component functions have continuous partial derivatives. We say that f is an immersion if Dsf is injective for all v in U and a submersion if Dof is surjective for allv in U. (a) Suppose that f:U-R" is an immersion. Prove that, for each v in U, we can find an open set V of U containing v, an open set W of R" containing f...
Real analysis subject 6. Prove the following slight generalization of the Mean Value Theorem: if f is continuous and differentiable on (a, b) and limy a f(v) and limyb- f(s) exist, then there is some z in (a, b) such that -a (Your proof might begin: "This is a trivial consequence of the Mean Value Theorem because ...".) .. 6. Prove the following slight generalization of the Mean Value Theorem: if f is continuous and differentiable on (a, b) and...
Question 1 (15 marks) Let f be a holomorphic, non-constant function on a domain 2 c C, and choose any open set UC 2. Define another set V CC as V = {f(u): u EU}. In other words, V is the image of U under the map f. Prove that V is open. Some hints: (a) consider an arbitrary point to EV, as well as a neighbourhood of that point; (b) consider a point u EU such that f(u) =...
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...
th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) Prove or find a counter example for the statement "a somewhat continuous func- (52) Let EC Rm, f: ER, and a e E. Prove or disprove that f is continuous at a there is a 6 0 such that whenever z,y D and la -vl <6, then f()-)e es tion is continuous." if and only if given any...
Please answer this question Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. 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)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...
Please prove Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function f: (a, b)- (c, d) for which f(a)-c and f(b)-d and show this is a homeomorphism.) Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function...