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 +g and fg are continuous at x0. (ii) cf is continuous at x0 for any constant c. (iii) If g(x0)̸=0, then f /g (defined on eD={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 +g is continuous at x0. □ Theorem 3.3.4 Let f : D→Rand let g: E→Rwith f (D) ⊂E. If f is continuous at x0 and g is continuous at f (x0), theng◦ 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, on R. (b) f (x)=x2 −3onR. (c) f (x)=|x|, on R. (d) f (x)=√x on[0,∞). (e) f (x)= 1 x onR\{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.
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