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 sequential characterization of uniform continuity. f(x)=1/(x^2-4), D=(2,4] hint:Considering sequences that converge to 2 f(x)=1/sqt(x) , D=(0,3] hint:Considering sequences that converge to 0

Proof f is uniformly continuous

(See attached file for full problem description with proper equations) --- 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.

Prove f is differentiable.

(See attached file for full problem description)

Proofs involving mean value theorem

(See attached file for full problem description)

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.