Examples and solution on continuitiy in real analysis
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A function won't have a minimum value on an open interval (a,b)(a,b) if it is, roughly speaking, constantly decreasing as it gets really close to the endpoints of the interval (and does not reach a minimum somewhere in the middle of the interval).
Take the function f(x)=−x2f(x)=−x2. On (−1/2,1/2)(−1/2,1/2) it has no minimum but reaches a maximum of 00 at x=0x=0. If you want a function that works on (0,1)(0,1), simply shift over ff to the right by 1/21/2 a unit, to obtain your final function:
−(x−1/2)2.
Take the function f(x)=−x2f(x)=−x2. On (−1/2,1/2)(−1/2,1/2) it has no minimum but reaches a maximum of 00 at x=0x=0. If you want a function that works on (0,1)(0,1), simply shift over ff to the right by 1/21/2 a unit, to obtain your final function:
−(x−1/2)2.
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Step-by-step explanation:
answer just represents on number line
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