Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Prove that for each non-closed subset of |R, there exists a continuous function defined on this set, which is not uniformly continuous.
If p is the point belonging to the closure of the set but not to the set itself, put f(x) = log(|x-p|).
page 185 in GOLDEN GATE
Prove that for each non-closed subset of |R, there exists a bounded continuous function, which has no maximum.
If p is the point belonging to the closure of the set but not to the set itself, put f(x)=1/(1+|x-p|).
(1) g : (0,oo) -> |R is diffable
(2) lim(x->oo) g(x)+g'(x) = A; A:-|R*
What can we conclude from this?
(3) lim(x->oo) g(x) = A
Hint: use the exponential function and L'Hospital's Rule.
page 190 in golden gate
Let f be a real-valued function defined on an interval. We informally say that f is convex iff for every two points x,y belonging to the domain of the function the segment joining points (x,f(x)) and (y,(f(y)) lies above the graph of the function.
Provide two formalizations of this idea.
(1) for every x,y, every a,b>=0 such that a+b=1
f(ax+by)<=af(x) + bf(y)
(2) for every distinct x,y, for every x<c<y,
f(c) <= ((f(y)-f(x))/(y-x)) * (c-x) + f(x)
f : I -> R, I is an interval on the real line
(1) for every x,y, every a,b>=0 such that a+b=1
f(ax+by)<=af(x) + bf(y)
(2) for every distinct x,y, for every x<=c<=y or y<=c<=x
f(c) <= ((f(y)-f(x))/(y-x)) * (c-x) + f(x)
Hint:
a = (y-c)/(y-x)
b = (c-x)/(y-x)
f:I->|R; I is an interval on the real line
(1) /\(x,y:-I) /\(a,b>=0, a+b=1) f(ax+by) <= af(x) + bf(y)
(2) /\(x<c<y :- I) f(c) <= ((f(y)-f(x))/(y-x))*(c-x) + f(x)
Prove that (2) => (1)
page 103 in palace
f : I -> R, I is an interval on the real line
(1) for every x,y, every a,b>=0 such that a+b=1
f(ax+by)<=af(x) + bf(y)
(2) for every x<y, for every x<c<y
(f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c)
Hint:
a,b>=0, a+b=1,
c = ax + by,
x<=c<=y,
a = (y-c)/(y-x)
b = (c-x)/(y-x)
(1) J is an open and connected subset of |R
(2) f:J->|R is a convex function
Does f have to be continuous?
YES.
page 115 in palace
f:A->B; g:B->R; A,B are intervals on the real line.
f,g are convex and g in increasing
Does g(f(x)) have to be convex?
YES.
Let I be an interval on the real line. Let f : I -> |R.
(1) /\x,y:-I f(x/2 + y/2) <= f(x)/2 + f(y)/2
(2) f is continuous
Does f have to be convex?
Yes.
Show that for every natural n, for all natural p,q such that p+q=2^n,
f(p/2^n * x + q/2^n * y) <= p/2^n * f(x) + q/2^n * f(y).
Then show that an appropriate set is dense and then use continuity to conclude the thesis.