2. Part (c) of Theorem 2.7.25 in the lecture notes told us that if f: R" → Rm is continuous, then for any closed set CCRm, f-1(C) is closed. With this in mind, for each of the following statements, either prove the statement or provide a counterexample. When providing a counterexample, you can choose your m and n. If you are proving a statement, you should do this in full generality; arbitrary m and n.: (a) If ƒ : Rn → Rm is a function so that f(C) is closed for any closed set CCR", then f is continuous. (b) If ƒ : R¹ → Rm is a function so that f(C) is closed for any closed set CCR, then f(0) is open for any open set OCRn. (c) If f: R → Rm is continuous, then f(C) is closed for any closed set CCR.

1. Suppose f: A → B is an injective function. Prove that f-1(f(C)) = C for all CCA. 2. Suppose A, B are sets, f : A → B is a function, and D₁, D₂ C B. In lectures we noted that the inverse image is well behaved with respect to set operations: f-¹ (D₁ D₂) = f¯¹ (D₁) n f¯¹ (D₂), f-¹ (D₁U D₂) = f¯¹ (D₁) Uf-¹ (D₂), and f¯-¹ (D₁ A D₂) = f¯¹ (D₁) A f¯¹(D₂). The image is not so well behaved. Yet one of the above three identities continues to hold with f replacing f-1 (and with D₁, D₂ replaced by subsets of A). Which is it? Can you prove this identity?

can you help wthi 5 please

Please do Exercise 17.5.19  part A and B and please show step by step and explain