75 This suggests that δ =ε/3 will be a good choice for δ. We write now a formal proof. Given ε >0, choose δ =ε/3. If |x−x0| <δ, we have | f (x)−f (x0)| =|(3x+7)−(3x0 +7)| =|3(x−x0)| =3|x−x0| <3δ =3( ε 3 ) =ε This shows that f is continuous at x0. Figure 3.1: Definition of continuity. Remark 3.3.1 Note that the above definition of continuity does not mention limits. This allows to include in the definition, points x0 ∈Dwhich are not limit points of D. If x0 is an isolated point of D, then there is δ >0 such that (x0−δ,x0+δ)∩D={x0}. It follows that for x ∈(x0−δ,x0+δ)∩D, | f (x)−f (x0)| =0<ε for any ε. Therefore, every function is continuous at an isolated point of its domain. To study continuity at limit points of D, we have the following theorem which follows directly from the definitions of continuity and limit. Theorem 3.3.1 Let f : D→Rand let x0 ∈Dbe a limit point of D. The following are equivalent: (i) f is continuous at x0. (ii) limx→x0 f (x) =f (x0). ■ Example 3.3.2 Let f : R→Rbe given by f (x) =3x 2 −2x+1. Fix x 0 ∈R. Since, from the results of the previous theorem, we have lim x→x0 f (x) = lim x→x0 (3x2 −2x+1) =3x2 0 −2x0 +1=f (x0). It follows that f is continuous at x0. The following theorem follows directly from the definition of continuity, Theorem 3.1.2 and Theorem 3.3.1 and we leave its proof as an exercise. Theorem 3.3.2 Let f : D→Rand let x0 ∈D. Then f is continuous at x0 if and only if for any sequence {xn}inDthat converges to x0, the sequence {f (xn)}converges to f (x0). The proofs of the next two theorems are straightforward using Theorem 3.3.2.
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