Please answer question 3.21 and 3.22. Thanks!
3.21 Prove the following facts: a) For fixed ECR, the function d(x, E) is continuous. b) If E and F are subsets of R, then d(E,F) = inf{dy, F): y EE}. c) If ACE and BCF, then d(E,F) <d(A,B). d) d(E,F) = d(E,F). 3.22 Prove the following facts: a) Suppose that F is a closed subset of R, K is a compact subset of R, and FOK = 0. Then d(F,K) > 0....
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9) Prove that if the function f is continuous on a, b, then there is c E [a, b such that f(x)dax a Ja f(e)
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9)...
9. Prove that the function f(x) = ax+b is uniformly continuous on R by directly applying the e, 8 definition of uniform continuity.
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
A polar curve r = f() has parametric equations x = f(0) cos(8), y = f(0) sin(8). Then, dy f() cos(0) + f (0) sin(e) d/ where / --f(8) sin(0) + / (8) cos(8) do Use this formula to find the equation in rectangular coordinates of the tangent line to r = 4 cos(30) at 0 = (Use symbolic notation and fractions where needed.)
9. If In E (a, b) is Cauchy, f : (a,b) + R is uniformly continuous, prove that f(x) is Cauchy.
Prove that there is no continuous bijection from the unit circle S1 = y21 onto any subset of R. (x,y) E R2
Prove that there is no continuous bijection from the unit circle S1 = y21 onto any subset of R. (x,y) E R2
1. Suppose m,b,c E R. Prove: f(1) = mx + b is continuous at c. 2. Prove: f(x) = x3 is continuous at 5.
Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for
all
x, y ∈ R. Prove that for some constant a ∈ R, f(x) = ax.
Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all X, Y E R. Prove that for some constant a ER, f(x) = ax.
Vector Analysis question
B.) (2pts) Let f: ECR” → R be defined as where x = (21, 22, ..., In). (i) Describe the domain E of f. Use the set notation. (Hint: the denominator can't be zero.) af Ij (ii) Show that at dr;= 713 (iii) Show that Vf = f2.