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Lime,
You are right that a function continuous only on (0,1) need not be bounded, but that is not what this problem is asking.
This particular problem requires that f be continuous on all of R. In other words, it must be continuous EVERYWHERE! Your function does not satisfy this requirement, so the result is not true for your function.
Instead take a function that is DEFINED and CONTINUOUS at every number. Then the result is true by the several proofs given in this thread.
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