MthT 430 Projects Chap 7b - Three Doubtful Theorems if Numbers = Q
MthT 430 Projects Chap 7b - Three Doubtful Theorems if Numbers = Q
While working on Chap7bproj, assume that we are in the
rational world - the only numbers available are the rational
numbers Q - there are no irrational numbers!
We could still define limits and continuity for functions, e.g.,
e- d Definition. The function f is continuous at a means: For every e > 0, there is some d > 0 such that, for all rational x, if |x -a| < d, then |f(x) - f(a)| < e.1We could also define continuity on intervals (of course the end
points of closed intervals [a,b] are rational).
We investigate the validity of the Three Hard Theorems in this
context.
For these projects, assume that we are in the rational
world . You are allowed to construct functions by formulas so
that for all rational numbers, x, f(x) is a rational number.
The usual polynomial and rational functions with integer or
rational coefficients will be continuous at all points where we do
not try to divide by 0. It is a deep result trigonometric
functions, etc., are not allowed.
Continuous Functions on
Intervals Have the Intermediate Value Property Doubtful Theorem 1. If f is continuous on [a,b] and f(a) < 0 < f(b), then there is some x in [a,b] such that f(x) = 0.
1.
Construct a function f on [0,1] such that
·
f is continuous on [0,1],
·
f(0) < 0 < f(1),
·
There is no x Î [0,1] such that f(x) = 0.
Hint: Use a variation of x ® x2 - 2.
Continuous Functions on Closed Intervals are
Bounded Doubtful Theorem 2. If f is continuous on [a,b], then f is bounded above on [a,b], that is, there is some number N such that f(x) £ N for all x in [a,b].
2.
Construct a function f on [0,1] such that
·
f is continuous on [0,1],
·
f(0) < 0 < f(1),
·
f is not bounded on [0,1].
Continuous Functions on Closed Intervals assume a
Maximum Value for the Interval Doubtful Theorem 3. If f is continuous on [a,b], then there is a number y in [a,b] such that f(y) ³ f(x) for all x in [a,b].
3.
Construct a function f on [0,1] such that
·
f is continuous on [0,1],
·
f is bounded on [0,1],
·
There is no number y in [0,1] such that f(y) ³ f(x) for all x in [0,1].
4.
Construct a function f on [0,1] such that
·
f is continuous on [0,1],
·
f(0) < 0 < f(1),
·
There is no number y in [0,1] such that f(y) ³ f(x) for all x in [0,1],
·
There is no number y in [0,1] such that f(y) £ f(x) for all x in [0,1].