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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

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.

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 \ 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

(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!).

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

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

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

(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.

Prove that x <= ax+by <= y.

very easy