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Does a horizontal line count as bounded? Also, how is $y=2^x$ bounded?

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    Don't forget (as I did at first reading) that "bounded" can mean "bounded below" (as is the case for $y = 2^x$).2012-09-11
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    A horizontal line is the graph of a bounded function: there are values that $y$ will never exceed and values it will never fall below. However, a horizontal line is not a bounded set: there is no distance that is not exceeded by that between some pair of points on the line.2012-09-11
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    Bounded usually does not mean "bounded either above or below". That's why we have the separate phrases "bounded", "bounded below" and "bounded above". We usually say that a function is bounded if and only if it is bounded above and bounded below.2012-09-11

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