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42 Cards in this Set

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continuous
if E, E' are ms and f:E->E' is a fct, then f is said to be continuous on E or more briefly, continuous, if f is continuous at all points of E
continuous(long def)
E and E' are ms with dists denoted d and d' and let p subzero elemt of E. Then f is said to be continuous at p subzero if given any real number epsilon greater than zero, there exists a real number delta greater than zero s.t. if p elemt of E and d(p, p subzero) < delta, then d'(f(p), f(p subzero) < epsilon
proposition on open subsets U of E'
Let E, E' be ms and f:E->E' a fct. Then f is continuous iff for every open subset U of E', the inverse image f^-1(U)={p elemt of E: f(p) elemt of U} is an open subset of E.
Corollary about continuous real valued fct
if f is a continuous real valued fct on the ms E, then for any a elemt of R {p elemt of E: f(p)>a} and {p elemt of E: f(p)< a} are open subsets of E

for the sets {x elemt R: x>a} and {x elemt R: x<a} are open subsets of R.
proposition g o f
Let E, E', E'' be ms, f: E->E', g: E->E" fcts. Then if f and g are continuous, so is the fct g o f:E->E". If p subzero elemt E and f is continuous at p subzero and g is continuous at f(p subzero) elemt of E', then g o f is continuous at p subzero.
Proposition f and g are real valued fcts
Let f and g be real valued fcts on ms E. If f and g are continuous at a point p subzero elemt E, then so are the fcts f+g, f-g, fg, f/g the last under the rule that g(p subzero) is not equal to 0 in which case g(p) is not equal to zero for all points p in some open ball of center p subzero.
Corollary p subzero be a cluster pt of ms E and let f and g be real valued fcts on c{p subzero}
p subzero be a cluster pt of ms E and let f and g be real valued fcts on c{p subzero} s.t. lim as p approaches p subzero f(p) and lim as p approaches p subzero g(p) exist. Then their sum, diff, product and quotient of their limits equal to the lim of their parts.
lemma x sub i: E to the n->R
For each i=1,2,...,n, the fct x sub i: E to the n->R defined by x sub i ((a1, a2,..., a subn)=a sub i is continuous
proposition x sub i o f
let E be a ms and f:E->E' a continuous fct. Then f is continuous at a point p subzero elemt E iff only each component fct x sub i o f, ..., x sub n o f is continuous at p subzero.
Theorem
Let E, E' be ms, f:E->E' a continuous fct. Then if E is compact so is its image f(E).
corollary 1 compact...bounded
let E, E' be ms, f:E->E' a continuous fct. Then if E is compact, f is bounded
corollary 2 max and min
A continuous real valued fct on a nonempty compact ms attains a max at some point, and also attains a min at some point.
Theorem compact....unifly cont
Let E, E' be ms and f:E->E' a continuous fct. If E is compact, then f is unformly continuous.
Defn of unifly cont
let E, E' be ms w/dists denoted d and d' and let f:E->E' be a fct. Then f is said to be unifly cont if given any real number epsilon greater than zero, there exists a real number delta greater than zero s.t. if p,q elemt E and d(p,q)< delta, then d'(f(p),f(q))< epsilon
Theorem connected...f(E) connected
let E, E' be ms, f:E->E' a cont fct. Then if E is connected, so its image f(E).
Corollary Intermediate value theorem
if a,b elemt of R, a<b, and f is cont real valued fcts on the closed interval [a,b], then for any real number gamma between f(a) and f(b), there exists at least one point c elemt (a,b) s.t. f(c)= gamma.
proposition nonempty intersection...connected
let {S sub i} for i elemt of I be a collection of connected subsets of ms E. Suppose there exists i subzero elemt I s.t. for each i elemt I the sets S sub i and S i subzero have a nonempty intersection. Then union of S sub i for i elemt of I is connected.
Theorem R...connected
R or any open or closed interval in R is connected.
epsilon-delta defn
f:(E,d)->(E', d'), p sub zero elemt of E, f is cont at p subzero if given any real value epsilon >0, there exists delta >0 s.t. p elemt of E and d(p, p subzero) <delta, then d'(f(p), f(p subzero))< epsilon or f(B(p subzero), delta) is contained in B(f(p subzero, epsilon).
sequential defn
f:(E,d)->(E', d'), p subzero elemt of E, f is ocntinuous at p subzero, if f(p sub n)->f(p subzero) whenever p sub n-> p subzero. f is ocntinuous if it continuous at every p elemt E.
Defn cauchy seq
A sequence of points p1, p2,p3,... in a ms is a cauchy seq if given any real number epsilon >0, there exists a positive integer N s.t. d(p sub n, p sub m)<epsilon whenever m,n >N.
Proposition conv seq
A convergent sequence of points in a ms is a Cauchy seq.
Proposition subseq of a Cauchy
Any subseq of a Cauchy sequence is a Cauchy sequence
Proposition Cauchy...bounded
A Cauchy sequence of points in a ms is bounded.
Proposition Cauchy seq that has a convergent subseq
A Cauchy sequence tht has a ocnvergent subsequence is itself convergent.
Defn completeness
A ms E is complete if every Cauchy sequence of points of E converges to a point of E.
Proposition closed subset of a complete
A closed subset of a complete ms is a complete ms.
Theorem R is...
R is complete.
Corollary E to the n
For any positive integer n, E to the n is complete.
Defn compactness
A subset S of a ms E is ocmpact if whenever S is ocntained in the union of a collection of open subsets of E, then S is contained in the union of a finite number of these open subsets.
proposition closed subset of a compact ms
Any closed subset of a compact ms is compact.
proposition compact...bounded
A compact subset of a ms is bounded. A compact ms is bounded.
proposition nested set property
Let S1, S2, S3,...be a sequence of nonempty closed subsets of a compact ms w/the property that S1 contains S2 contains... There exists at least one point that belongs to each of the set S1, S2, S3,...
defn cluster point
If E is a ms, S is subset of E, and p a point of E, then p is a cluster point of S if any open ball w/center p contains an infinite number of points of S.
Theorem infinite subset
An infinite subset of a compact ms has at least one cluster point.
Corollary 1 compact...con seq
Any sequence of points in a compact ms has a convergent subsequence.
Corollary 2 compact...complete
A compact ms is complete.
Corollary 3 compact...closed
A compact subset of a ms is closed.
Lemma bounded subset of E to the n...union of finite
Let S be a bounded subset of E to the n. Then for any epsilon>0, S is contained in the union of a finite number of closed balls of radius epsilon.
Theorem closed bounded subset
Any closed bounded subset of E to the n is compact.
Defn connected
A ms E is ocnnected if the only subsets of E which are both open and closed are E and null set. A subset S of a ms is a connected subset if the subspace S is connected.
Proposition subset of R...not connected
any subset of R which contains two distinct points a and b and does not contain all points between a and b is not connected.