Is a horizontal line an increasing or decreasing function?

Question

This is the definition of an increasing and decreasing function.

"A function $f(x)$ increases on an interval I if $f(b)≥f(a)\;\;\forall b>a$, where $a,b \in I$. If $f(b)>f(a) \;\;\forall \;\;b>a$, the function is said to be strictly increasing.

Conversely, A function $f(x)$ decreases on an interval I if $f(b)≤f(a)\;\;\forall b>a$, where $a,b \in I$. If $f(b)<f(a) \;\;\forall \;\;b>a$, the function is said to be strictly decreasing.

Then how would a horizontal line be described? If $f(x_2)=f(x_1)\;\; \forall x_2>x_1$ does that mean that a horizontal line meets the definition of an increasing function and a decreasing function?

this means the function is constant – user860374 Mar 19 '16 at 13:59 — user860374, Mar 19 '16 at 13:59
Yes, a constant function is both increasing and decreasing. – Git Gud Mar 19 '16 at 14:04 — Git Gud, Mar 19 '16 at 14:04

SchrodingersCat · Accepted Answer · 2016-03-19T14:03:13.760

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It means it is neither strictly increasing nor strictly decreasing, it is a constant function.

However, the way you have defined increasing and decreasing functions, the given horizontal line satisfies both definitions.

But it is better to avoid putting this type of function in any such class. It should be rightly referred to as a constant function.

edited Mar 19 '16 at 14:03

answered Mar 19 '16 at 13:59

SchrodingersCat

24,472

Of course it is a constant function. But on his definitions (which are fairly standard) it is also both an increasing function and a decreasing function, but neither a strictly increasing function nor a strictly decreasing function. – almagest Mar 19 '16 at 14:00
Got it. Thanks. – Jonathan Mar 19 '16 at 14:02
@Jonathan Downvotes?? – SchrodingersCat Mar 19 '16 at 14:05
I didn't downvote you. – Jonathan Mar 19 '16 at 14:05
Why down votes!? Please have a kind discussion instead. :-) – Mikasa Mar 19 '16 at 14:06

score 0 · Answer 2 · answered Mar 19 '16 at 14:01

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Yes, an horizontal line is a function $f$ such that $f(x)=c$ for all $x\in \mathbb{R}$, and, if $a\leq b\to f(a)=c\leq c =f(b)$. Idem for $\geq$

answered Mar 19 '16 at 14:01

Martín Vacas Vignolo

4,710

score 0 · Answer 3 · answered Mar 19 '16 at 14:11

Inject $f(x)=c$ in the definitions:

"A function $c$ increases on an interval I if $c≥c\;\;\forall b>a$, where $a,b \in I$. If $c>c \;\;\forall \;\;b>a$, the function is said to be strictly increasing.

Conversely, A function $c$ decreases on an interval I if $c≤c\;\;\forall b>a$, where $a,b \in I$. If $c<c \;\;\forall \;\;b>a$, the function is said to be strictly decreasing."

You should be able to see for yourself what property holds.

Is a horizontal line an increasing or decreasing function?

3 Answers3