(Sequential Criterion for Continuity) Let (S₁, d₁) and (S2, d2) be metric spaces, and f: S₁ S₂ be a function. Show that f is continuous at the point c E S₁ if and only if for every sequence (xn) in S₁ that converges to c, the sequence f(xn) converges to f(c). (Please use the correct definitions to prove this theorem.)

Class: Mathematical Analysis/Real Analysis; C2

Advanced Calculus: Suppose f and g are differentiable functions on an interval I. Show that iff′(x) = g′(x) for all x ∈ I and f(x0) = g(x0) for some x0 ∈ I, then f(x) = g(x) forall x ∈ I.

2. Is the proof correct? Either state that it is, or circle the first error and explain what is incorrect about it. If the proof is not correct, can it be fixed to prove the claim true? Claim: If f : (0,1] → R and g : [1,2) → R are uniformly continuous on their domains, and f(1) = g(1), then the function h : (0, 2) → R, defined by h(æ) = { F(x) for x € (0, 1] | 9(x) for x e [1, 2) is uniformly continuous on (0, 2). Proof: Let e > 0. Since f is uniformly continuous on (0, 1], there exists d1 > 0 such that if x, y E (0, 1] and |æ – y| 0 such that if æ, y E [1, 2) and |r – y| < d2, then |9(x) – 9(y)| < €/2. Let 8 = min{d1, &2}. Now suppose r, Y E (0, 2) with x < y and |x – y| < 8. If x, y E (0, 1], then |x – y| < 8 < 81 and so |h(x) – h(y)| = |f (x) – f(y)|< e/2 < e. If x, y € (1,2), then |æ – y| < 8 < d2 and so |h(x) – h(y)| = |9(x) – g(y)| < e/2 < e. If x € (0,1) and y E (1,2), then |x – 1| < |x – y| < 8 < dị and |1 – y| < |æ – y| < 8 < 82 so that |h(x) – h(y)| = |f (x) – f(1) + g(1) –…