110年 - 110 國立中央大學_碩士班招生考試_數學系/數學組(一般生):高等微積分#105339

(1.2) (2 points) If are bounded sequences in R, then

(1.4) (2 points) If A is connected inRⁿ, thenA is also connected.

(1.6) (2 points) Let f be a continuous real function on R and let Z(f) be the set of all at which f(p) = 0. Then Z(f) is closed.

(1.7) (2 points) The function is uniformly continuous on (0, 1).

(1.8) (2 points) Let f be a real function on [0,2], then f is Riemann integrable on [0,2] if and only if|f| is Riemann integrable on [0, 2].

(1.10) (2 points) Let a uniformly convergent sequence of polynomials on [0,2] and . Then f is differentiable.

(2) (10 points) Let B = f{(x, y). Show that B is open set.

(3) (15 points) Let (M,d) be metric space and M. Show that the set {x₁, x₂, x₃,...] is compact set.

(4) (20 points) Every rational t can be written in the form, where n> 0, and m and n are integers without any common divisors. When a = O, we take n = 1. Consider the function f defined on R by
 Prove that f is continuous at every irrational point, and that f has a discontinuity at every rational point.