Unformatted text preview:  x  → ∞ , for large values of  x  the function sin x(1+ x 2 )1 is positive near the points π/ 2+2 nπ and is negative near the pointsπ/ 2 + 2 nπ . By the intermediate value theorem, there is a solution in each interval [π/ 2 + 2 nπ,π/ 2 + 2 nπ ], where n = 1 , 2 ,... 5. continuous and unbounded on [0 , 1); YES e.g. f ( x ) = 1 / (1x ). 6. continuous and unbounded on [0 , 1]; NO, the range of a continuous function on a closed bounded interval is a closed bounded interval. 7. continuous on [0 , 1] and { f ( x ) : x ∈ [0 , 1] } = (0 , 1); NO, for the same reason 8. continuous on (0 , 1) and { f ( x ) : x ∈ (0 , 1) } = [0 , 1]; YES, e.g. f ( x ) = sin(4 πx ). 9. continuous on (0 , 1) and { f ( x ) : x ∈ (0 , 1) } = [0 , 1] ∪ [3 , 4]. NO, by the intermediate value theorem. 1...
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 Spring '09
 Topology, Intermediate Value Theorem, Limits, Continuous function, Metric space

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