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I came across two definitions which are as follows:

First definition:

A function f is increasing at a point $x_0$ if there is a δ > 0 such that $f(x)⩽f(x_0)⩽f(y)\,\forall x∈(x_0−δ,x_0)\cap D_f$ and $\forall y∈(x_0,x_0+δ)\cap D_f$

Second definition:

A function f is increasing at a point $x_0$ if there is a δ > 0 such that $f(x)<f(x_0)<f(y)\,\forall x∈(x_0−δ,x_0)\cap D_f$ and $\forall y∈(x_0,x_0+δ)\cap D_f$

As per first definition constant function is increasing at every point but as per second it is not which one is correct definition?

Or should we define two terms increasing at point and strictly increasing at point as we use to do for intervals?

One more doubt at end point of domain can we discuss f is increasing/decreasing. I think we can discuss as in this case we will only take one side of neighborhood.

Makar
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  • Authors vary. Some authors use increasing and strictly increasing, other authors use non-decreasing and increasing, still other authors use non-decreasing and strictly increasing, yet still other authors use weakly increasing and strongly increasing, etc. This is just one of those annoying things that you have to look out for because there is no universal standard, similar to $\subset$ (does this mean subset or proper subset?) and natural number (is $0$ included or is $0$ not included?). – Dave L. Renfro Apr 15 '18 at 11:45
  • @DaveL.Renfro So what is your suggestion – Makar Apr 15 '18 at 16:50
  • Is the following conclusion true:Increasing at point:

    A function f is increasing at a point $x_0$ if there is a δ > 0 such that $f(x)⩽f(x_0)⩽f(y),\forall x∈(x_0−δ,x_0)\cap D_f$ and $\forall y∈(x_0,x_0+δ)\cap D_f.$

    Strictly increasing at point:

    A function f is strictly increasing at a point $x_0$ if there is a δ > 0 such that $f(x)<f(x_0)<f(y),\forall x∈(x_0−δ,x_0)\cap D_f$ and $\forall y∈(x_0,x_0+δ)\cap D_f.$

     – Makar Apr 15 '18 at 17:22
  • Yes, this looks correct. See the Stack Exchange question Can a function be increasing at a point? and A handful of theorems about point increase by Tadeusz Janaszak (2013; .pdf file were "Janaszak_A_handful_of_theorems_about_point.pdf 261,95 KB" appears) and (continued) – Dave L. Renfro Apr 15 '18 at 20:34
  • Paramanand Singh's blog entry Monotone Functions: Part 1 and p. 160 of More applications of full covering by Karen Klaimon (1990). – Dave L. Renfro Apr 15 '18 at 20:39
  • @DaveL.Renfro thanks a lot. Your references and comments answered my problem. Thank you very much once again. – Makar Apr 16 '18 at 15:50

1 Answers1

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The first definition is increasing function. The second one is strictly increasing function and both are the correct definitions. However, a constant function falls in increasing functions not in strictly increasing functions. That is my opinion.

WKhan
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  • Please read carefully I want definition for increasing at a point not over an interval. – Makar Apr 15 '18 at 09:22
  • taking derivative can be helpful like a function will be increasing if $f'(0)>0$. Otherwise you should check monotonically increasing/decreasing function. Probably that will answer your question. Thanks – WKhan Apr 15 '18 at 09:35
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    The OP's question has to do with pointwise notions of increasing, not interval notions of increasing. Also, no assumption on differentiability (or even continuity) is being made. Finally, even when differentiability is assumed, the assumption $f'(0) > 0$ is stronger than increasing at a $x=0.$ For example, $f(x) = x^3$ is strictly increasing at $x=0,$ but $f'(0) = 0.$ – Dave L. Renfro Apr 15 '18 at 11:41