76 3.3 Continuity Theorem 3.3.3 Let f ,g: D → R and let x0 ∈ D. Suppose f and g are 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 (defned on De= {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 in D that converges to x0. Since f and g are 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 ( f + g)(xn)= lim ( f (xn)+ g(xn)) = lim f (xn)+ lim g(xn)= f (x0)+ g(x0)=( f + g)(x0). n→∞ n→∞ n→∞ n→∞ 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 → R and let g: E → R with f (D) ⊂ E. If f is continuous at x0 and g is continuous at f (x0), then g◦ f is continuous at x0. Exercises 3.3.1 Prove, using defnition 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. x + 3 (c) f (x)= , x0 = 1. x + 1 3.3.2 Prove, using defnition 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 − 3 on R. (c) f (x)= |x|, on R. √ (d) f (x)= x on [0,∞). 1 (e) f (x)= on R \{0}. x 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 in R. 1 xsin , if x ̸ = 0; (a) f (x)= x 0, if x = 0. sinx , if x ≠ 0; (b) f (x)= x 1, if x = 0.
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