a) Show that f is discontinuous at any x 6=
0.
b) Show that f is continuous at x = 0.
c) Show that f is differentiable at x = 0 and compute
the value f 0 (0).
d) Show that f is not integrable on the interval [1, 2]
(or any interval, but I don’t mind if you use that interval
specifically).
(x2 (x EQ) f(x)=o (x &Q)
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
Answer C
6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x.
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...
Problem 2. Let C[0, 1] be the set of all continuous functions from [0, 1] to R. For any f, g є Cl0, 11 define - max f(x) - g(z) and di(f,g)-If(x) - g(x)d. a) Prove that for any n 2 1, one can find n points in C[O, 1 such that, in daup metric, the distance between any two points is equai to 1. b) Can one find 100 points in C[0, 1] such that, in di metric, the...
I do not need the two metrics to be proved (that they are a
metric).
Problem 2. Let C[0, 1] be the set of all continuous functions from [0, 1] to R. For any f, g є Cl0, 11 define - max f(x) - g(z) and di(f,g)-If(x) - g(x)d. a) Prove that for any n 2 1, one can find n points in C[O, 1 such that, in daup metric, the distance between any two points is equai to 1....
(1) If f: R₃ R a continuous function such that f(x)² > 0 for all xER. Show that either f(x) >0 for all a ER or f(x) <0 all X E R.
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an
Suppose f is a continuous and differentiable function on...
5. Is f continuous at f(1)? (10 points) [-x2 +1, 4x, f(x) = -5, -1<x<0 0<x<1 x=1 1<x<3 3<x<5 - 4x + 8 1,