77 (c) f (x) = sinx |x| , if x̸ =0; 1, if x =0. (d) f (x) = cos πx 2 , if |x| ≤1; |x−1|, if |x| >1. (e) f (x) =lim n→∞ sin π 2(1+x2n) , x ∈R. 3.3.4 Let f : R→Rbe the function given by f (x) =( x2 +a, if x >2; ax−1, if x ≤2. Find the value of asuch 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. (a) f (x) =(1, if x ∈Q; −1, if x ∈Qc. (b) f (x) =( x, if x ∈Q; 0, if x ∈Qc. (c) f (x) =( x, if x ∈Q; 1−x, if x ∈Qc. 3.3.6 ▶Let g,h: R→Rbe continuous functions and define f (x) =( g(x), if x ∈Q; 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 defined on Rthat is continuous at eachxi, i =1, . . . ,k, and discontinuous at all other points. 3.3.8 Let f : D→Rand 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 Dand when it is not.) 3.3.10 Suppose that f,gare continuous functions onRand 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.
RkJQdWJsaXNoZXIy NTc4NTAz