a)
use the sequential definition of continuity to prove that f(x)=|x|
is continuous.
b) theorem 17.3 states that if f is continuous at x0, then |f|
is continuous at x0. is the converse true? if so, prove it. if not
find a counterexample.
hint: use counterexample
Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous
bijection. Prove that if (X, dX ) is compact, then f is a
homeomorphism. (Hint: it might be convenient to use that a function
is continuous if and only if the inverse image of every open set is
open, if and only if the inverse image of every closed set is
closed).
Differential Geometry (Mixed Use of Vector Calculus & Linear
Algebra)
1A. Prove that if p=(x,y) is in the set where y<x and if
r=distance from p to the line y=x then the ball about p of radius r
does not intersect with the line y=x.
1B. Prove that the set where y<c is an open set.
Proof of If and Only if (IFF) and
Contrapositive
Let x,y be integers. Prove that the product xy is odd if and
only if x and y are both odd integers.
Proof by Contradiction
Use proof by contradiction to show that the difference of any
irrational number and any rational number is irrational. In other
words, prove that if a is irrational and b is a rational numbers,
then a−b is irrational.
Direct Proof
Using a direct proof, prove that:...
1. Use the ε-δ definition of continuity to prove that (a) f(x) =
x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at
every x0 not equal to 0.
3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is
discontinuous at every x0 not equal to 0. (b) Is f continuous at x0
= 0 ? Give an answer and then prove it.
4. Let f and g...
a. Let f be a real function. Prove that f is convex iff −f is
concave.
b. Let f and g be real functions. Prove that if f and g are
convex, then f + g is convex