A- Show that if a function is continuous on all of R and equal to 0 at every rational point then it must be identically 0 on all of R
b- if f and g are continuous on all of R and f(r)=g(r) at every rational point,must f and g be the same function?

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a. Proof:
For any irrational point x, we can always find a sequence of rational points x_n such that x_n->x as n->oo. From the definition, ...

Solution Summary

There are two proofs regarding rational points and continuity of functions.

Determine where the function f(x)= x + [|x^2|] - [|x|] is continuous.
I think the correct answer is that the function is continuous for its domain but not defined at x=0.
Can someone explain this problem to me and help me understand the greatest integer and absolute value functions?
keywords: continuity

Let be a compact interval and let A be a collection of continuousfunctions on which satisfy the properties of the Stone-Weierstrass Theorem
[Stone-Weierstrass Theorem: Let K be a compact subset of and let A be a collection of continuousfunctions on K to R with the properties:
a) The constant function belongs to A.
b) I

A). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints).
Let p(x,y) = max_[0,1]|x'(t) - y'(t)|.
1).Show that ( M,p) fails to be a metric space.
2). Let p(x,y) = |x(0) - y(0)| + max_[0,1]|x'(t) - y'(t)|. Is (M,p) now a metric space?
Please

3. a) Let M be a connected topological space and let f : M ---> R be continuous. Pick m1,m2 2 M and suppose that f(m1) < f(m2). Let x 2 R be such that f(m1) < x < f(m2). Show that there is m M with f(m) = x. (Hint: Use a connectedness argument.)
b) Give R1 the usual product topology as the product of infinite copies of the rea

Question 1.
1) Suppose (V, | * |) is a normed space. If x, y E V and r is a positive real number, show that the open r-balls Br(x) and Br(x + y) in V are homeomorphic.
2) Suppose V and W are two normed spaces. If A : V ---> W is a linear map, then show that it is continuous at every point v E V if and only if it is continuou

Detailed step by step calculations of the attached questions regarding complex variables including the domain, limits and continuity of complex functions.
1. For each of the functions below, describe the domain of definition, and write each function in the form f(z) = u(x,y) +iv(x,y)
1) f(z) = z^2 / (z+z)
2) f(z) = z^3
3