Home / Questions / Consider two properties that a function f might satisfy:
(look at attachment as you will see the question more clear) 1st property (A) for all strictly positive real numbers, e, there exists a strictly positive real number, d, such that, ?x,y in the domain of f, v(x-y)^2 < d ? v (f (x) - f (y))^2 < e 2nd property (B) v(f (x) - f (y))^2 = c v (x -y)^2 a) Show that v (x-y)^2 = |x-y|, where |z| is the absolute value of z. Provide an example to show that v (x-y)^2 is ? x-y. b) Does f(x) = vx, whose domain is the non-negative reals, satisfy: i. property A? ii. property B? c) Provide an example of a function that satisfies: i. properties A and B. ii. neither property A nor B.
3. Consider two properties that a function f might satisfy: (A) for all strictly positive real numbers, e, there exists a strictly positive real number, d, such that, Vx,y in the domain off, (B) Vx,y in the domain ofl, there exists a finite constant c ~ 0 such that Ju (x) - f {y))2 ::;cJ{x _ y)2 (a) Show that Jex - y)2 = Ix - yl, where Izl is the absolute value of z. Provide an example to show that J(x - y)2 =1= x - y. (Hint: it will later be useful to remember that [z]< c forc~ 0 is equivalent to -c < x < c.) (b) Doesf (x) = ...;x, whose domain is the non-negative reals, satisfy: i. property A? ii. property B? (c) Provide an example of a function that satisfies: i. properties A and B. ii. neither property A nor B. ..,
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