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Math Help - Increasing and Strictly increasing

  1. #1
    Senior Member pankaj's Avatar
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    Increasing and Strictly increasing

    A function is said to be increasing on [a,b] if for every x_{1} ,x_{2}\in(a,b) where x_{2} > x_{1}then f(x_{2}) > f(x_{1})

    I read in a book that function y = f(x) is increasing on [a,b] if f'(x)\geq 0 on the interval (a,b)(but f'(x) should not become zero on any sub-interval of (a,b))

    This is given in standard texts

    So what exactly does strictly increasing mean i.e when is a function said to be strictly increasing on the interval [a,b].

    The definition of strictly increasing was given as follows:

    The function y = f(x) is strictly increasing on [a,b] if f'(x)> 0 on the interval (a,b)

    A confusion gets created with regard to use of \geqsign and > sign


    So what exactly is the definition of increasing function and what exactly is the definition of strictly increasing function
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  2. #2
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    Quote Originally Posted by pankaj View Post
    A function is said to be increasing on [a,b] if for every x_{1} ,x_{2}\in(a,b) where x_{2} > x_{1}then f(x_{2}) > f(x_{1})

    I read in a book that function y = f(x) is increasing on [a,b] if f'(x)\geq 0 on the interval (a,b)(but f'(x) should not become zero on any sub-interval of (a,b))

    This is given in standard texts

    So what exactly does strictly increasing mean i.e when is a function said to be strictly increasing on the interval [a,b].

    The definition of strictly increasing was given as follows:

    The function y = f(x) is strictly increasing on [a,b] if f'(x)> 0 on the interval (a,b)

    A confusion gets created with regard to use of \geqsign and > sign


    So what exactly is the definition of increasing function and what exactly is the definition of strictly increasing function
    Monotonic function - Wikipedia, the free encyclopedia
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  3. #3
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    Quote Originally Posted by pankaj View Post
    A function is said to be increasing on [a,b] if for every x_{1} ,x_{2}\in(a,b) where x_{2} > x_{1}then f(x_{2}) > f(x_{1})

    I read in a book that function y = f(x) is increasing on [a,b] if f'(x)\geq 0 on the interval (a,b)(but f'(x) should not become zero on any sub-interval of (a,b))

    This is given in standard texts

    So what exactly does strictly increasing mean i.e when is a function said to be strictly increasing on the interval [a,b].

    The definition of strictly increasing was given as follows:

    The function y = f(x) is strictly increasing on [a,b] if f'(x)> 0 on the interval (a,b)

    A confusion gets created with regard to use of \geqsign and > sign


    So what exactly is the definition of increasing function and what exactly is the definition of strictly increasing function
    f is an increasing function if and only if a> b implies f(a)\ge f(b).

    f is a strictly increasing function if and only if a> b implies f(a)> f(b.

    Notice that those definitions do NOT require the function to be differentiable and the first allows the function to be constant on an interval. Indeed, a constant function is "increasing" in that sense.

    Note: some textbooks use "increasing" in the sense of "strictly increasing" here and use "non-decreasing" in the sense of "increasing" here.
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  4. #4
    Senior Member pankaj's Avatar
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    This is what was confusing me
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