Increasing and decreasing functions

The idea of increasing or decreasing functions is very intuitive, although one must be able to formulate it mathematically. The following graphs show it:

Spontaneously one would say that the first graph corresponds to an increasing function, while the second one corresponds to a decreasing function.

Now consider the following graphs:

In this example one can still tell what function is increasing and which one is decreasing. However we can also see that both functions have parts where they are not increasing or decreasing.

These four graphs are representative of four different kinds of functions that will be properly defined.

Strictly increasing function (Graph 1)

A function is strictly increasing at any point (that we call $a$) when the following property is satisfied:

Function $f$ is strictly increasing in $a⟺\mathrm{\exists }E(a)$ such that $\mathrm{\forall }x\in E(a)$ we have: $$\begin{array}{rcl}x>a& ⟹& f(x)>f(a)\\ x<a& ⟹& f(x)<f(a)\end{array}$$

Which is read as follows: The function $f$ is strictly increasing in $a$ if, and only if, there is an environment of $a$ so that for any $x$ that belongs to this environment we have that if $x$ it is strictly greater than $a$ then $f(x)$ is strictly greater than $f(a)$, and if $x$ it is strictly less than $a$ then $f(x)$ is less than $f(a)$.

Clearly the idea of strictly increasing is simpler than its formal definition.

Strictly decreasing function (Graph 2)

A function is strictly decreasing at any point $a$ when the following property is satisfied:

Function $f$ is strictly decreasing in $a⟺\mathrm{\exists }E(a)$ such that $\mathrm{\forall }x\in E(a)$ we have: $$\begin{array}{rcl}x>a& ⟹& f(x)<f(a)\\ x<a& ⟹& f(x)>f(a)\end{array}$$

Which is read as follows: The function $f$ is strictly decreasing in $a$ if, and only if, there is an environment of $a$ so that for any $x$ that belongs to this environment we have that if $x$ is strictly greater than $a$ then $f(x)$ is strictly less than $f(a)$, and if $x$ is strictly less than $a$ then $f(x)$ is greater than $f(a)$.

Increasing function (Graph 3)

The two situations seen up to now are clearly the most restrictive cases.

The third and fourth graphs do not satisfy these definitions since there are points where the function is neither increasing nor decreasing. For that we need the definition of an increasing or decreasing function (without being strictly increasing or decreasing).

A function is increasing at any point (that we call $a$) when the following property is satisfied:

Function $f$ is increasing in $a⟺\mathrm{\exists }E(a)$ such that $\mathrm{\forall }x\in E(a)$ we have: $$\begin{array}{rcl}x<a& ⟹& f(x)\ge f(a)\\ x<a& ⟹& f(x)\le f(a)\end{array}$$

Which is read as follows: The function $f$ is increasing in $a$ if, and only if, there is an environment of $a$ so that for any $x$ that belongs to this environment we have that if $x$ is greater than $a$ then $f(x)$ is greater or equal to $f(a)$, and if $x$ is less than $a$ then $f(x)$ is less or equal to $f(a)$.

Clearly the idea of strictly increasing is simpler than its formal definition.

Decreasing function (Graph 4)

We can thus, guess the definition of a decreasing function:

A function is decreasing at any point $a$ when the following property is satisfied:

Function $f$ is decreasing in $a⟺\mathrm{\exists }E(a)$ such that $\mathrm{\forall }x\in E(a)$ we have: $$\begin{array}{rcl}x>a& ⟹& f(x)\le f(a)\\ x<a& ⟹& f(x)\ge f(a)\end{array}$$

Which is read as follows: The function $f$ is decreasing in $a$ if, and only if, there is an environment of $a$ so that for any $x$ that belongs to this environment we have that if $x$ is greater than $a$ then $f(x)$ is less or equal to $f(a)$, and if $x$ is strictly less than $a$ then $f(x)$ is greater or equal to $f(a)$.

Fortunately a brief way of writing these definitions exists using the concept of derivative.

• Increasing at $a$: $f^{\prime }(a)\ge 0$
• Decreasing at $a$: $f^{\prime }(a)\le 0$

(Or its most restrictive version for strictly increasing and strictly decreaseing, with the symbols $>$ and $<$).