Results 1 to 4 of 4

Thread: Increasing and Strictly increasing

  1. #1
    Senior Member pankaj's Avatar
    Joined
    Jul 2008
    From
    New Delhi(India)
    Posts
    318

    Increasing and Strictly increasing

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

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

    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 $\displaystyle [a,b]$.

    The definition of strictly increasing was given as follows:

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

    A confusion gets created with regard to use of $\displaystyle \geq$sign and $\displaystyle >$ sign


    So what exactly is the definition of increasing function and what exactly is the definition of strictly increasing function
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    12,880
    Thanks
    1946
    Quote Originally Posted by pankaj View Post
    A function is said to be increasing on $\displaystyle [a,b]$ if for every $\displaystyle x_{1} ,x_{2}\in(a,b)$ where $\displaystyle x_{2} > x_{1}$then $\displaystyle f(x_{2}) > f(x_{1})$

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

    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 $\displaystyle [a,b]$.

    The definition of strictly increasing was given as follows:

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

    A confusion gets created with regard to use of $\displaystyle \geq$sign and $\displaystyle >$ 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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,769
    Thanks
    3027
    Quote Originally Posted by pankaj View Post
    A function is said to be increasing on $\displaystyle [a,b]$ if for every $\displaystyle x_{1} ,x_{2}\in(a,b)$ where $\displaystyle x_{2} > x_{1}$then $\displaystyle f(x_{2}) > f(x_{1})$

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

    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 $\displaystyle [a,b]$.

    The definition of strictly increasing was given as follows:

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

    A confusion gets created with regard to use of $\displaystyle \geq$sign and $\displaystyle >$ 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 $\displaystyle f(a)\ge f(b)$.

    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member pankaj's Avatar
    Joined
    Jul 2008
    From
    New Delhi(India)
    Posts
    318
    This is what was confusing me
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. increasing functions prove strictly increasing
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 14th 2010, 03:19 PM
  2. Prove that f is strictly increasing
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Jan 18th 2010, 08:29 AM
  3. strictly increasing function
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Mar 29th 2009, 01:40 AM
  4. Prove that it's strictly increasing
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: Apr 28th 2008, 05:54 PM
  5. strictly increasing function
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Nov 23rd 2006, 10:40 AM

Search Tags


/mathhelpforum @mathhelpforum