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12. Let a and b be real number, a < b, and let f be a real-valued function that is defined on the interval (a,b). Which of the following statements implies that f is continuous on (a,b)?
I. The range of f is an interval.
II. The graph of f has a highest and a lowest point.
III. The graph of f intersects any horizontal line at most once.
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A. None |
B. I only |
C. II only |
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D. III only |
E. The correct answer is not given by A, B, C, or D. |
Solution: The answer is A
Consider the following two real-valued discontinuous functions, f(x) and g(x), that are defined on the interval (0,2), and their graphs:
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Statement I: Does not imply that f(x) is continuous on (0,2) since the range of f(x) (a discontinuous function) is an interval.
Statement II: Does not imply that f(x) or g(x) are continuous on (0,2) since the graph of both discontinuous functions has a highest and a lowest point.
Statement III: Does not imply that g(x) is continuous on (0,2) since the graph of g(x) intersects any horizontal line at most once.
Hence, none of the statements are true.
