Final Exam Definitions

Question 0

infimum

Answer 0

Let S be a nonempty set of real numbers. If S is bounded below, then a number w is said to be an infimum (or a greatest lower bound) of S if:

(1) w is a lower bound of S
(2) if t is any lower bound, the w >= t

Question 1

supremum

Answer 1

Let S be a nonempty subset of the real numbers. If S is bounded above, then a number u is said to be a supremum (or least upper bound) of S if it satisfies the conditions:

(1) u is an upper bound of S
(2) if t is any upper bound of S, then u <= t

Question 2

convergence of a sequence

Answer 2

A sequence X = (xn) in the real numbers is said to converge to a real number x, or x is said to be a limit of (xn), if for every epsilon > 0 there exists a natural number K such that for all n >= K, the terms of xn satisfy |xn-x| < epsilon

Question 3

Cauchy Sequence

Answer 3

A sequence X = (xn) of real numbers is said to be a Cauchy sequence if for every epsilon > 0 there exists a natural number H such that for all natural numbers n,m >= H, the terms xn, xm satisfy |xn-xm| < epsilon

Question 4

limit of a function

Answer 4

Let A be a subset of the real numbers, and let c be a cluster point of A. For a function f from A to the real numbers, a real number L is said to be a limit of f at c if, given any epsilon > 0, there exists a delta > 0 such that if x is in A and 0 < |x-c| < delta, then |f(x) - L| < epsilon

Question 5

continuous function

Answer 5

Let A be a subset of real numbers, let f be a function from A to the real numbers, and let c be in A. We say that f is continuous at c if, given any number epsilon > 0, there exists delta > 0 such that if x is any point of A satisfying |x-c| < delta, then |f(x) - f(c)| < epsilon

Question 6

uniformly continuous function

Answer 6

Let A be a subset of the real numbers and let f be a function from A to the real numbers. We say that f is uniformly continuous on A if for each epsilon > 0 there exists a delta > 0 such that if x and u are in A and |x-u| < delta, then |f(x) - f(u)| < epsilon

Question 7

derivative

Answer 7

Let I be a subset of the real numbers and an interval, let f be a function from I to the real numbers, and let c be in I. We say that a real number L is the derivative of f at c if given any epsilon > 0 there exists a delta > 0 such that if x is in I and 0 < |x-c| < delta, then |(f(x)-f(c))/(x-c) - L| < epsilon or f’(c) = limit as x goes to c of f(x)-f(c)/(x-c)

Question 8

Riemann sum

Answer 8

We define the Riemann sum of a function f from [a,b] to the real numbers corresponding to the tagged partition P to be the number
S(f;P) := the sum from i=1 to n of f(ti)(xi-x(i-1))

Question 9

Riemann integrable

Answer 9

A function f from [a,b] to the real numbers is said to be Riemann integrable on [a,b] if there exists a real number L such that for every epsilon > 0 there exists delta > 0 such that if P is any tagged partition of [a,b] with ||P|| < delta, then |S(f; P) - L| < delta

Question 10

step function

Answer 10

The function f from [a,b] to the real numbers is a step function if f takes on only finitely many values