(1) Given a continuous function f, show that raC Hint: parts. (1) Given a continuous function f, show that raC Hint: parts.
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...
1. Show that f : (R,Te) → (R,Tj.), given by f(x)-z?, is a continuous bijection whose inverse function is not continuous. Here Tee and Tie are the countable complement and finite complement topologies respectively
Let f [a, b [a, b] such Exercise 28: R be continuous. Show that there exists x E : that 1 f(x)= f. 6-a Hint: Intermediate value theorem
Let f [a, b [a, b] such Exercise 28: R be continuous. Show that there exists x E : that 1 f(x)= f. 6-a Hint: Intermediate value theorem
9. Show that the function f() is continuous at 4.
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,...
Show that the function
is continuous at point 1, by using
f(:0) = +2 4.72 – 3 9-3
Is the function given by f(x) = continuous at x = 5? Why or why not? 5x+2, for x 55, 5* 5x - 18, for x>5, Choose the correct answer below. O A. The given function is continuous at x = 5 because the limit is 3. OB. The given function is continuous at x = 5 because lim f(x) does not exist. X-5 OC. The given function is not continuous at x = 5 because f(5) does not exist....
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...
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
(1 point)Several Values of a continuous function f(x) are given
below:
(1 point) Several values of a continuous function f(x) are given below: 2= f(x) = 1 9 2 -12 3 5 4 5 Find (Si (f (t)) dt)|g_1 = | Note: If it is not possible to find an answer from the given information then enter DNE to indicate that the answer Does Not Exist.