77 sinx , if x ≠ 0; (c) f (x)= |x| 1, if x = 0. πx cos , if |x|≤ 1; (d) f (x)= 2 |x − 1|, if |x| > 1. π (e) f (x)= lim sin , x ∈ R. n→∞ 2(1 + x2n) 3.3.4 Let f : R → R be the function given by ( x2 + a, if x > 2; f (x)= ax − 1, if x ≤ 2. Find the value of a such that f is continuous. 3.3.5 Determine the values of x at which each function is continuous. The domain of all the functions is R. (1, if x ∈ Q; (a) f (x)= −1, if x ∈ Qc . ( x, if x ∈ Q; (b) f (x)= 0, if x ∈ Qc . ( x, if x ∈ Q; (c) f (x)= 1 − x, if x ∈ Qc . 3.3.6 ▶ Let g,h : R → R be continuous functions and defne ( g(x), if x ∈ Q; f (x)= h(x), if x ∈ Qc . Prove that if g(a)= h(a), for some a ∈ R, then f is continuous at a. 3.3.7 ▷ Consider k distinct points x1,x2,...,xk ∈ R, k ≥ 1. Find a function defned on R that is continuous at each xi, i = 1,...,k, and discontinuous at all other points. 3.3.8 Let f : D → R and let x0 ∈ D. Prove that if f is continuous at x0, then | f | is continuous at this point. Is the converse true in general? 3.3.9 Prove Theorem 3.3.2. (Hint: treat separately the cases when x0 is a limit point of D and when it is not.) 3.3.10 Suppose that f ,g are continuous functions on R and f (x)= g(x) for all x ∈ Q. Prove that f (x)= g(x) for all x ∈ R. 3.3.11 Prove parts (ii) and (iii) of Theorem 3.3.3.
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