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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.

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State a condition which, in complete metric spaces, is equivalent to the existence of the limit of a function at a point. This is analogous to the Cauchy criterion for the limit of a sequence in a complete metric space.
(We're talking about the limit of function f at point x0.)
For every positive e, there is a positive b such that for every x,y satisfying d(x,x0)<b and d(y,x0)<b, it follows that g(f(x),f(y))<e.
page 157 in 1st analysis
Consider a real-valued function defined on a bounded open interval. If this function is bounded and continuous, does it have to be uniformly continuous?
NO. sin(1/x) defined on (0,1)
If it were uniformly continuous, it could be extended to a continuous function on [0,1]. This is impossible because the limit of sin(1/x) as x approaches zero doesn't exist. Hence this function is not uniformly continuous.
A, B, C are sets.
A \ (B \ C) = ?
A \ (B \ C) = (A \ B) u (A n C)
Let f be a real-valued function defined on a connected subset of |R. Suppose that f is differentiable. What is the relation between the derivative of f and whether f satisfies a Lipschitz condition? Give the answer without proof.
For every real K, these two conditions are equivalent.
(1) |f'(x)| <= K, for all x
(2) |f(x)-f(y)| <= K|x-y|, for all x,y
(The proof requires Lagrange's mean-value theorem.)
page 156 in golden gate
Prove that for each natural number n,
e - (1/0! + 1/1! + 1/2! + 1/3! + ... + 1/n!) < 1/(n * n!).
page 79 in OLDTIMER
page 176 in the first notebook for analysis
Prove that the number e is irrational.
Hint: use the fact that for each natural number n,
e - (1/0! + 1/1! + 1/2! + 1/3! + ... + 1/n!) < 1/(n * n!).
page 175 in the first notebook for analysis
What is the formula for the n-th derivative of the product of two functions? (Supposing that both the functions have the n-th derivative.)
the sum from k=0 to k=n
(n) (n-k) (k)
(k)f g
+(k=0 to k=n) Newton(n,k) * f^(n-k) * g^(k)
page 171 in the palace notebook
What is the formula for the n-th derivative of sin?
sin(n)(x) = sin(x+n*pi/2)
What is the formula for the n-th derivative of cos?
cos(n)(x) = cos(x+n*pi/2)
a,b >= 0 and a+b=1, x<=y
Prove that x <= ax+by <= y.
very easy

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