Term
f is continuous at c if... 

Definition
limit as x goes to c of f(x) = f(c). Which also means that for each epsilon > 0, there is some delta > 0 so that if x is an element of A and xc < delta, then f(x)f(c) < epsilon. 


Term
Boundedness of Continuous Functions 

Definition
If f : I maps to R is continuous on the closed interval I = [a,b], then f is bounded on I. 


Term

Definition
Let f : I goes to R be continuous on the closed interval I = [a,b] and let m = inf { f(x):x element of I } and M = sup { f(x):x element of I }. Then there must be values of c and d in I so that f(c)=m and f(d)=M. 


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Intermediate Value Theorem 

Definition
Let f : I goes to R be continuous on the closed interval I = [a,b] and let f(a) != f(b). For any value t between f(a) and f(b), there is a value c element (a,b) so that f(c) = t. 


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Definition
If f is continuous on the closed interval [a,b], and f(a)f(b) < 0, then there is some point c element of (a,b), so that f(c)=0. 


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The function f is uniformly continuous on the set S if... 

Definition
for every epsilon > 0, there is a delta > 0 so that for all x,u where xu < delta, we have f(x)f(u) < epsilon. 


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I is an interval of real numbers, and let c be an interior point of I. f is differentiable at c if... 

Definition
lim as x goes to c of [f(x)f(c)/(xc)] exists. If it does exist, then the value of the limit is called the derivative of f at c and is denoted by the symbol f'(c). 


Term
f has a relative minimum at c element of I if... 

Definition
there is a neighborhood U of c such that f(x) >= f(c) for all x element of U intersect I. 


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f has a relative maximum at c element of I if... 

Definition
there is a neighborhood U of c such that f(x) <= f(c) for all x element of U intersect I. 


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Definition
If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b) = 0, then there is some value c element of (a,b) so that f'(c) = 0. 


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Definition
If f is continuous on [a,b] and differentiable on (a,b) then there is at least one value c element of (a,b) such that f'(c) = [(f(b)f(a)/(ba)]. 


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Corollary 20.6 If f has a bounded derivative on an interval I, then... 

Definition
f is uniformly continuous on I. 


Term
If f(x) is integrable on [a,b], then a theorem states that 

Definition
For all epsilon > 0, there exists a partition of [a,b] s.t. U(f,P)L(f,P) < epsilon 


Term
If f'(x) is bounded for all x element of I, then 

Definition
f is uniformly continuous on I 


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If for all x element of S, f(x) does not equal the supremum of all the function values, then 

Definition


Term
To see if somethings integrable: (4 steps) 

Definition
1. FInd partitions 2. Find U and L 3. Find inf U and sup L 4. Does inf U = sup L? if it does, integrable! 


Term
If F(x) equals (integral)(a to x) f(t) dt, assuming that f(t) is integrable on [a,b], then 

Definition
F(x) is a continuous function F'(x) = f(x) at any point where f is continuous. 


Term
If f is bounded and continuous, 

Definition
then f is uniformly continuous on S 


Term
If f is unbounded on S, then 

Definition
f is not uniformly continuous on S, unless S = [a,infinity), then f(x)=x would be a counterexample 


Term
If f has an unbounded derivative on S, 

Definition
then f is not uniformly continuous on S 


Term
If f has a bounded derivative on S, 

Definition
then f is uniformly continuous on S. 


Term
If f is unbounded and f' is unbounded on S, then 

Definition
f is not uniformly continuous on S. 


Term
If f is continuous on [a,b], then 

Definition
f is uniformly continuous on [a,b] 


Term
If f is continuous on (a,b), then 

Definition
f is not necessarily uniformly continuous on (a,b), for example f(x) = 1/x 


Term
If f is continuous on [a,infinity), then 

Definition
f is not necessarily uniformly continuous, f(x) = x^{2} on [1,infinity) is a counterexample. 


Term
If if is bounded and continuous on [a,b] 

Definition
then f is uniformly continuous on [a,b] 


Term
if f is bounded and continuous on (a,b) 

Definition
then f is not necessarily uniformly continuous on (a,b), f(x) = sin(1/x) on (0,1) is a counterexample. 


Term
If f is bounded and continuous on [a,infinity) 

Definition
then f is not necessarily uniformly continuous on [a, infinity), f(x) = sin(x^{2}) from [1,infinity) is a good counterexample 


Term
If f is unbounded on [a,b], 

Definition
then f is not uniformly continuous on [a,b] 


Term
If f is unbounded on (a,b), 

Definition
then f is not uniformly continuous on (a,b). 


Term
If f is unbounded on [a,infinity), 

Definition
then f could be uniformly continuous, like with f(x) = x. 


Term
If f' is unbounded on [a,b], then 

Definition
f could be uniformly continuous on [a,b], like with f(x) equals square root of x on [0,1]. 


Term
If f' is unbounded on (a,b), then 

Definition
f could be uniformly continuous on (a,b), like f(x)=square root of x on (0,1) 


Term

Definition
then f is certainly uniformly continuous on S, (any type of interval). 


Term
If f and f' are unbounded on [a,b] or (a,b), 

Definition
then f is not uniformly continuous on S. 


Term
If f is integrable on [a,b], and F is continuous on [a,b] with F'=f on (a,b), then 

Definition
integral from a to b of f(x) dx = F(b)F(a). 


Term

Definition
1. Denial
2. Construct sequence and subsequence
3. Say lim x_{nk} = x_{0}
4. Conclude f(x_{nk}) converges to f(x_{0}). 


Term

Definition
1. Say {y_{n}} goes to M
2. Say f(x_{n}) = y_{n}
3. Say {x_{nk} goes to d.}
4. So {f(x_{nk})} goes to M and f(d) = M. 


Term
Continuous function is uniformly continuous 

Definition
1. Denial
2. Construct {x_{nk}} that goes to x_{0}
3. Assert that {u_{nk}} goes to x_{0}
4. Then u_{nk } x_{nk} + x_{nk}  x_{0} < e/2 + e/2 = e
5. And f(x_{nk})  f(u_{nk}) < e_{0} 


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Definition
1. There exists an x so f(x) > 0
2. Form a bounded set S.
3. Let M = sup of S
4. By Lemma 20.1, there is a c s.t. f(c) = M
5. Therefore f'(c) = 0. 


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Definition
1. Apply Rolle's Theorem to weird function.
2. Check for cont. and diff.
3. Does f(a) = f(b) = 0?
4. Then differentiate weird function.
5. Show that f'(c) = f(b)f(a)/(ba). 


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Definition
1. L(f,P) < sup L(f,P) < inf U(f,P) < U(f,P) for any partition.
2. Since there is a partition so UL < e, it is also true that 0<supinf<e.
3. Since e is arbitrary, lower sum equals upper sum. 


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Definition
1. Apply MVT to F on [x_{i1},x_{i}]
2. Rearrange and take summation.
3. Show that left equals F(x_{n})F(x_{0}) = F(b)F(a) 


Term
Fundamental Theorem of Calculus (Part 1) 

Definition
1. L < integral < U
2. Then L < summation (Lemma 27.1) < U
3. Then L < F(b)F(a) (Lemma 27.1) < U
4. Since UL is less than epsilon, integral  (F(b)F(a)) < epsilon.
5. So, integral = F(b)F(a). 


Term
Fundamental Theorem of Calculus (Part 2.1) 

Definition
1. There is a B s.t. f(x) < B.
2. Assume x > c
3. Do some work and F(x)F(c) < B(xc)
4. Let delta = epsilon/B
5. So F(x) is cont. at c. 


Term
Fundamental Theorem of Calculus (Part 2.2) 

Definition
1. MVT with F,x,x_{0}
2. Do simplification (add in a f(x_{0}) + f(x_{0}))
3. Show that left limit thing is zero.
4. So F'(x) = f(x). 

