74 3.3 Continuity 3.2.5 ▶ Let f : D → R and let x0 be a limit point of D. Suppose that | f (x) − f (y)|≤ k|x − y| for all x,y ∈ D \{x0}, where k ≥ 0 is a constant. Prove that limx →x0 f (x) exists. 3.2.6 Prove Theorem 3.2.3 3.2.7 Determine the one-sided limits limx →3+[x] and limx →3− [x], where [x] denotes the greatest integer that is less than or equal to x. 3.2.8 Find each of the following limits if they exist: x + 1 (a) lim , x→1+ x − 1 (b) lim x3 sin(1/x) , x→0+ (c) lim(x − [x]). x→1 3.2.9 For a ∈ R, let f be the function given by ( x2 , if x > 1; f (x)= ax − 1, if x ≤ 1. Find the value of a such that limx →1 f (x) exists. 3.2.10 Determine all values of x0 such that the limit limx →x0 (1+ x − [x]) exists. 3.2.11 Let a,b ∈ R and suppose f : (a,b) → R is increasing. Prove the following: (a) If f is bounded above, then limx →b− f (x) exists and is a real number. (b) If f is not bounded above, then limx →b− f (x)= ∞. State and prove analogous results in case f is bounded below and in case that the domain of f is one of (−∞,b), (a,∞), or (−∞,∞). 3.3 Continuity Defnition 3.3.1 Let D be a nonempty subset of R and let f : D → R be a function. The function f is said to be continuous at x0 ∈ D if for any real number ε > 0, there exists δ > 0 such that | f (x) − f (x0)| < ε for all x ∈ D with |x − x0| < δ . If f is continuous at every point x ∈ D, we say that f is continuous on D (or just continuous if no confusion occurs). ■ Example 3.3.1 Let f : R → R be given by f (x)= 3x + 7. Let x0 ∈ R and let ε > 0. We need to fnd a δ such that if |x − x0| < δ , then | f (x) − f (x0)| < ε. As we have done before with limits of functions, we start with | f (x) − f (x0)|. | f (x) − f (x0)| = |(3x+ 7) − (3x0 + 7)| = |3(x − x0)| = 3|x − x0|.
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