Functions of Bounded Variation and Rectifiable Curves
Functions of bounded variation
(a) fx x 2 sin1/x if x 0, f0 0. (b) fx x sin1/x if x 0, f0 0.
6.1 Determine which of the follwoing functions are of bounded variation on 0, 1 .
Proof: (a) Since f x 2x si
The Real And Complex Number Systems
1.1 Prove that there is no largest prime.
Proof : Suppose p is the largest prime. Then p! + 1 is NOT a prime. So,
there exists a prime q such that
q |p! + 1 q |1
which is impossible. So, there is no largest pri
Written by Men-Gen Tsai email: [email protected] 1. Prove that the empty set is a subset of every set. Proof: For any element x of the empty set, x is also an element of every set since x does not exist. Hence, the empty set is a subset
The Real and Complex Number Systems
Written by Men-Gen Tsai email: [email protected] 1. 2. 3. 4. 5. 6. Fix b > 1. (a) If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q , prove that (bm )1/n = (bp )1/q . Hence it makes sense to dene br = (bm )