3.2.5 Concept Check: Bounds on functions 1 Concept Check: Bounds on functions Let f : D →  be a function, and denote the range of f by f(D) . If f(D) is bounded above, then we say that f is bounded above and the supremum of f is the supremum of its range: sup f = sup f(D) . If the range of f has a maximum, then the maximum of f is the maximum of its range: max f = max f(D) . If f(D) is bounded below, then we say that f is bounded below and the infimum of f is the infimum of its range: inf f = inf f(D) . If the range of f has a minimum, then the minimum of f is the minimum of its range: min f = min f(D) . f is bounded if it is bounded above and below. 1. Sketch a graph of each of the following five functions: g : − {0} → 
g(x) = (sinx)/ x
α : (− π2 , π2 ) → 
β : (0,1) →  α (x) = tan(x)
β (x) = 1/ ln(x)
• Is the function bounded above? below? Is it bounded? • Does the function have a supremum? a maximum? • Does the function have an infimum? a minimum? Solution: The functions f, g, and h are bounded above, bounded below, and bounded. Each of them therefore has a supremum and an infimum by the Completeness Axiom. None of these three functions has a maximum, and only g has a minimum. is not bounded above or below so it does not have a supremum, infimum, maximum or minimum. is bounded above but not below. It has a supremum but not a maximum. Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810 3.2.5 Concept Check: Bounds on functions 2 2. True or False? a) If D ⊂  is bounded, then any function f : D→  is bounded. Solution: This is false. The functions and in Part (1) are counterexamples. b) If D ⊂  is unbounded, then any function f : D→  is unbounded. Solution: This is false. The functions f, g, and h in Part (1) are counterexamples. c) Given a domain D ⊂  , there exists a bounded function f : D→  . Solution: This is true. For example, a constant function on any domain is bounded. d) Given a domain D ⊂  , there exists an unbounded function f : D→  . Solution: This is false. On a domain consisting of only finitely many points, every function is bounded. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810