The definition of increasing function given in my school maths text book is
Let $I$ be an open interval contained in the domain of a real valued function $f$. Then $f$ is said to be increasing on $I$ if $a \lt b \implies f(a) \le f(b)$ for all $a, b \in I$.
And a theorem given after it is
$f$ is increasing in $I$ if $f'(x) \gt 0 \; \forall x \in I$
Shouldn't it be $f'(x) \ge 0$.
Is constant function an increasing function? Or a function like $f(x) = x^3$ where $f'(x) = 0$ at some or all points which are increasing according to definition.
I looked on wolfram and other sites. It is same there.
Edit:The definition of decreasing function given is
$f$ is decreasing on $I$ if $a \lt b \implies f(a) \ge f(b)$
A constant function follow this definition too.
$\endgroup$ 82 Answers
$\begingroup$To say that a function is increasing (decreasing) usually means that the the function doesn't decrease (doesn't increase). As Steven Gregory has already said, sometimes the word "monotone" is added for emphasis. When the function actually always increases (decreases) it is said to be strictly increasing (decreasing).
A function which is both increasing and decreasing in this sense never decreases and never increases, and so is constant.
In the "math for computer science" books, at least in the most popular ones, a different terminology is used. "Increasing" means "strictly increasing" and "monotone increasing" is rendered as "non-decreasing." (Similarly for decreasing functions.) Personally, I hate this terminology, but it doesn't really matter.
You seem to be bothered that the words "increasing" and "decreasing" are used in an unfamiliar sense. You'll just have to live with it, I'm afraid.
$\endgroup$ $\begingroup$Some would call $\forall x \in I, f'(x) \ge 0$ a non decreasing function.
Also some would call $\forall x \in I, f'(x)>0$ a strictly increasing function.
But, yes, $\forall x \in I, f'(x) \ge 0$ is often defined as an increasing function. In that case, the function $f(x)=1$ is an increasing function.
Sometime the word monotone gets thrown in there too.
You may as well get used to the fact that not all mathematicians use the same definition for things. As long as the book is consistent, that is acceptable.
INCREASING:
- $\forall x,y \in I, x < y \implies f(x) \le f(y)$
- $\forall x \in I, f'(x) \ge 0$
STRICTLY INCREASING:
- $\forall x,y \in I, x < y \implies f(x) < f(y)$
- $\forall x \in I, f'(x) > 0$
MOTONIC:
- Strictly increasing or strictly decreasing
