Continuous Functions and Open Intervals

Assume f is continuous on the interval (a,b) that contains x0 and f(x0)>0. Show there is an open interval J that contains x0 and m>0 such that f(x)>=m for each x in J.

Epsilon-Delta Definition of Continuity

Proof that f is continuous for each x in D in accordance with the epsilon-delta defitinition of continuity(can use the defintion involving f(x+h) (2 problems) f(x)=x/(x+1), D={x in R:x>-1} (can restrict |h|<(x+1)/2 f(x)=1/sqrt(x-4), D={x in R:x>4} (can restrict |h|<(x-4)/2

Proving that f is not uniformly continuous

The following theorem could be used to write the proof. A theorem states that if d:D-->R is uniformly continuous on D iff the following condition is satisfied: If un and vn are both sequences in D, then lim as n-->infinity (f(un)-f(vn))=0 Show f is not uniformly continuous on D making use of the sequent ...continues

Uniformly Continuous Functions and Mean Value Theorem

Assume that f is differentiable for each x and there exists M>0 such that for each x Prove that f is uniformly continuous on D. Hint: Can use the mean value theorem. keywords: differentiability, continuity

Continuity Proof

Assume that f(x) is continuous in some open interval J that contains the point a, f’(x) exists for each x and limit of f’(x) as xa exists. Prove that f is differentiable at a and f’(a)=limit of f’(x) as xa keywords: differentiability

Proof using Mean Value Theorem of Continuous, Increasing Functions

Assume that f and f ’ are continuous on [a,b], and f ’’(x) exists and f ’’(x)>0 for each x 1) Prove that f ’ is increasing on [a,b] Hint: the graph is concave up on this interval. 2) Prove that f(x) f(c) for each x if c and f ‘(c)=0.

Proof with integral

(See attached file for full problem description) --- Assume that f is continuous on [a,b] and f(x) 0 for each x [a,b]. Prove that >0 if there exists c (a,b) such that f(c)>0.

Differentiation of composite function - integral form

(See attached file for full problem description with proper symbols) --- Assume that f is continuous on [a,b], g is differentiable on [c,d], g([c,d]) [a,b] and F(x) = For each x [c,d]. Prove that F’(x)=f(g(x))g’(x) For each x (c,d).

Proof involving integral

(See attached file for full problem description with proper symbols) --- Assume that f is continuous on [a,b], g is differentiable on [c,d], g([c,d]) [a,b] and F(x) = For each x [c,d]. Prove that F’(x)=f(g(x))g’(x) For each x (c,d). ---

Subset of a countable set is countable

Prove that every subset S of a countable set X is itself countable.