# Increasing and Strictly increasing

• Jun 28th 2009, 11:41 PM
pankaj
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
• Jun 28th 2009, 11:55 PM
Prove It
Quote:

Originally Posted by pankaj
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
• Jun 29th 2009, 08:37 AM
HallsofIvy
Quote:

Originally Posted by pankaj
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.
• Jun 29th 2009, 09:09 AM
pankaj
This is what was confusing me