74 3.3 Continuity 3.2.5 ▶Let f : D→Rand 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: (a) lim x→1+ x+1 x−1 , (b) lim x→0+ x3sin(1/x) , (c) lim x→1 (x−[x]). 3.2.9 For a∈R, let f be the function given by f (x) =( x2, if x >1; ax−1, if x ≤1. Find the value of asuch 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∈Rand suppose f : (a,b) →Ris 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 Definition 3.3.1 Let Dbe a nonempty subset of Rand let f : D→Rbe a function. The function f is said to be continuous at x0 ∈Dif for any real number ε >0, there exists δ >0 such that | f (x)−f (x0)| <ε for all x ∈Dwith 0<|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→Rbe given by f (x) =3x+7. Let x0 ∈Rand let ε >0. We need to find 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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