Let (X,d) be a metric space. The graph of f : X → R is the...

Let (X,d) be a metric space. The graph of f : X → R is the set {(x, y) E X X Rly = f(x)}. If X is connected and f is continuous, prove that the graph of f is also connected.

Expert Solution

Colby Messinger answered 2 years ago

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset of R^k...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset...

Answer for a and be should be answered independently. Let (X,d) be a metric space, and...

Answer for a and be should be answered independently. Let (X,d) be a metric space, and a) let A ⊆ X. Let U be the set of isolated points of A. Prove that U is relatively open in A. b) let U and V be subsets of X. Prove that if U is both open and closed, and V is both open and closed, then U ∩ V is also both open and closed.

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that...

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that x is isolated if the set {x} is open in E. (a) Suppose that there exists r > 0 such that Br(x) contains only finitely many points. Prove that x is isolated. (b) Let E be any set, and define a metric d on E by setting d(x, y) = 0 if x = y, and d(x, y) = 1 if x and y...

1(i) Show, if (X, d) is a metric space, then d∗ : X × X →...

1(i) Show, if (X, d) is a metric space, then d∗ : X × X → [0,∞) defined by d∗(x, y) = d(x, y) /1 + d(x, y) is a metric on X. Feel free to use the fact: if a, b are nonnegative real numbers and a ≤ b, then a/1+a ≤ b/1+b . 1(ii) Suppose A ⊂ B ⊂ R. Show, if A is not empty and B is bounded below, then both inf(A) and inf(B) exist and...

Let X be a metric space and t: X to X be a map that preserves...

Let X be a metric space and t: X to X be a map that preserves distances: d(t(x), t(y)) = d(x, y). Give an example in whicht is not bijective. Could let t: x to x+1,x non-negative, but how does this mean t is not surjective? Any help will be much appreciated!

Let f(x) = 14 − 2x. (a) Sketch the region R under the graph of f...

Let f(x) = 14 − 2x. (a) Sketch the region R under the graph of f on the interval [0, 7]. The x y-coordinate plane is given. There is 1 line and a shaded region on the graph. The line enters the window at y = 13 on the positive y-axis, goes down and right, and exits the window at x = 6.5 on the positive x-axis. The region is below the line. The x y-coordinate plane is given. There...

Let X be a compact space and let Y be a Hausdorff space. Let f ∶...

Let X be a compact space and let Y be a Hausdorff space. Let f ∶ X → Y be continuous. Show that the image of any closed set in X under f must also be closed in Y .

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

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