Limits and Continuity

Question 1

The following are equivalent:

(a) f is continuous at c

Answer 1

(b) If x_n is any sequence in D such that x_n converges to c, then lim f(x_n)=f(c)
(c) For every neighborhood V of f(c) there exists a neighborhood U of c such that f(U intersect D) c V

ALSO, if c is in D’, then all are equivalent to

(d) f has a limit at c and lim f(x)=f(c)

Question 2

F is DISCONTINUOUS iff

Answer 2

there is a sequence x_n in D such that x_n converges to c BUT the sequence f(x_n) does not converge to f(c)

Question 3

Arithmetic of Continuity

f and g are continuous functions at c, then

Answer 3

(a) f + g and fg are continuous at c
(b) f/g is continuous at c if g(c) not = 0
(c) kf is continous at c

Question 4

f(D)cE, f is continuous at a point c in D and g is continuous at f(c)

Answer 4

then the composition g•​f is continuous at c

Question 5

Theorem 5.2.14 f is continuous on D iff for every open set G in R

Answer 5

there exists an open set H in R such that

(H intersect D) = f^-1(G)

Question 6

Corollary 5.2.15 f mapped R to R is continuous iff f^-1(G) is open

Answer 6

whenever G is open in R

Question 7

Theorem 5.3.2 Let D be a compact subset of R and suppose f : D→R is continuous.

Answer 7

Then f(D) is compact

Question 8

Corollary of Compact Subsets

Let D be a compact subset of R, and f:D is continuous

Answer 8

Then f assumes maximum and minimum values on D, that is f(x₁) f(x) f(x₂)

Question 9

Intermediate Value Theorem

Answer 9

Suppose f : [a,b] is continuous. Then f has the intermediate value property on [a,b].

That is: if k is between f(a) and f(b), then there is a c in (a,b) such that f(c)=k

Question 10

Theorem 5.3.10 Intervals

Let I be a compact inteveral and f: I is continuous

Answer 10

then the set f(I) is a compact interval

Question 11

Definition of Uniform Continuity

f is uniformly continuous on D if

Answer 11

for every E>0 there is σ>0 such that |f(x) - f(y)|<e>
</e>

Question 12

Theorem 5.4.6

If f is continous on a compact set D, then

Answer 12

f is uniformly continuous on D

Question 13

Let f be uniformly continuous on D and suppose x_n is Cauchy in D,

Answer 13

then f(x_n) is also Cauchy, that is convergent

Question 14

Extension of a continous function

Iff f is continuous on (a,b) and can be extended to be continuous on [a,b],

Answer 14

then f is uniformly continous

If f extedened is uniformly continous on [a,b], then f on (a,b) is uniformly continous

Question 15

Definition of Limit of Function

Let f : D → R and c is an accumulation point of D, a real number L is the limit of f at c if

Answer 15

for every E>0 there exists a σ>0 such that |f(x)-L| < E when x is in D and 0 < |x-c| < σ

Question 16

Neighborhood Definition of Limits

lim_x→cf(x) = L IFF

Answer 16

for every neighborhood V of L there is a deleted neighborhood U of c such that f(U) c V

Question 17

Sequential Criterion for Limits

f : D and c in D’

Then lim_x→cf(x) = L IFF

Answer 17

for every s_n→c in D with s_n not= c for all n, the sequence (f(s_n))→L

Question 18

Corollary of Seq Criterion

Answer 18

f can have only one limit at c

Question 19

Converse of Sequential Criterion

Iff f DOES NOT have a limit at c

Answer 19

there is an s_n in D where s_n not=c such that sn→c but (f(sn)) not convergent