76 3.3 Continuity Theorem 3.3.3 Let f,g: D→Rand let x0 ∈D. Suppose f andgare continuous at x0. Then (i) f +gand f gare continuous at x0. (ii) c f is continuous at x0 for any constant c. (iii) If g(x0)̸ =0, then f /g(defined on e D={x ∈D: g(x)̸ =0}) is continuous at x0. Proof: We prove (i) and leave the other parts as an exercise. We will use Theorem 3.3.2. Let {xn} be a sequence inDthat converges tox0. Since f andgare continuous at x0, by Theorem 3.3.2 we obtain that {f (xn)}converges to f (x0) and {g(xn)}converges to g(x0). By Theorem 2.2.1 (i),we get that {f (xn)+g(xn)}converges to f (x0)+g(x0). Therefore, lim n→∞ (f +g)(xn) =lim n→∞ (f (xn)+g(xn)) =lim n→∞ f (xn)+lim n→∞ g(xn) =f (x0)+g(x0) = (f +g)(x0). Since {xn}was arbitrary, using Theorem 3.3.2 again we conclude f +gis continuous at x0. □ Theorem 3.3.4 Let f : D→Rand let g: E→Rwith f (D) ⊂E. If f is continuous at x0 andg is continuous at f (x0), then g◦ f is continuous at x0. Exercises 3.3.1 Prove, using definition 3.3.1, that each of the following functions is continuous at the indicated point x0: (a) f (x) =3x−7, x0 =2. (b) f (x) =x2 +1, x 0 =3. (c) f (x) = x+3 x+1 , x0 =1. 3.3.2 Prove, using definition 3.3.1, that each of the following functions is continuous on the given domain: (a) f (x) =ax+b, a,b∈R, onR. (b) f (x) =x2 −3 onR. (c) f (x) =|x|, onR. (d) f (x) =√x on [0,∞). (e) f (x) = 1 x on R\ {0}. 3.3.3 Determine the values of x at which each function is continuous. The domain of all the functions is R. You may assume the functions sine and cosine are continuous inR. (a) f (x) = xsin 1 x , if x̸ =0; 0, if x =0. (b) f (x) = sinx x , if x̸ =0; 1, if x =0.
RkJQdWJsaXNoZXIy NTc4NTAz