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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: xE', 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(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) (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)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.