2. Suppose that f is continuous at every point of [a, b] and that f(x) =...
Suppose that f is continuous at every point of [a, b] and that f(x) = 0 whenever x is rational. Prove that f(x) = 0 for all x ∈ [a, b]. Suppose that f is continuous at every point of (a, b) and that f(x) = 0 whenever x is rational. Prove that f(x) = 0 for all x € [a, b].
HW #25 and suppose f is continuous on [a,b] f is diff. on (a, b) with fra) = f(b) = 0 that for every real number & prove there exists ceca, b) sit. f'is = dfrey (Hint Annly f'reo- &f(2)=0 Rolle to fixo e x ,
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on C if and only if Vf(x*)-(x - x*)2 0 for all x in C. Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on...
Can you help with this? Thank you always. Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that there is an interval 1 (x,-1/n, xo + 1 /n), where n is a natural number, such that f (x) >0 for all x in I. (Hint: Argue by contradiction.) Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that...
9. Suppose that f is bounded on [a, b], and f is continuous at each point of la, b] except for .in (a,b). Prove that f is integrable on [a, b).
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.
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...
6.59. Let f be a continuous function on [a, b]. Suppose that there exists a positive constant K such that If(x) <K for all x in [a, b]. Prove that f(x) = 0 for all x in [a, b]. *ſ isoidi,
Suppose that f : [a, b] → a, b] is continuous. Prove that f has a fixed point, i.e., prove that there exists ce [a, b] such that f(c) = c.
2. Let DCR, and suppose the functions f:D + R and g: D R are continuous at to ED. Use the - definition of continuity to prove that f+g and fg are both continuous at ro e D: that is, prove that for every e > 0 there exists 8 >0 such that (+9)(2)-(+9)(20) <and (9)(x) - (49)(20) << whenever re D and r-rol < 8. (Hint. Use inequalities similar to those we used to prove the cor- responding results...