Difference between increasing function and strictly increasing function in terms of derivatives? Consider the following theorem and remarks from the Application of Derivatives chapter in p. 201 of the NCERT textbook.

Theorem:
Let $f$ be continuous on $[a, b]$ and differentiable on the open
interval $(a,b)$. Then
(a) $f$ is increasing in $[a,b]$ if $f′(x) > 0$ for each $x \in (a,
 b)$
(b) $f$ is decreasing in $[a,b]$ if $f′(x) < 0$ for each $x \in (a,
 b)$
(c) $f$ is a constant function in $[a,b]$ if $f′(x) = 0$ for each $x
 \in (a, b)$
Remarks:
(i) $f$ is strictly increasing in $(a, b)$ if $f′(x) > 0$ for each $x
 \in (a, b)$
(ii) $f$ is strictly decreasing in $(a, b)$ if $f′(x) < 0$ for each $x
 \in (a, b)$
(iii) A function will be increasing (decreasing) in R if it is so in
every interval of R.

I am thinking that the theorem is incomplete and the remarks are correct. That theorem has to be as follows:

(a) $f$ is increasing in $[a,b]$ if $f′(x) \ge 0$ for each $x \in (a,
 b)$
(b) $f$ is decreasing in $[a,b]$ if $f′(x) \le 0$ for each $x \in (a,
 b)$

Since there is a difference between increasing function and strictly increasing function, I am feeling that the theorem given in the textbook is faulty. Am I correct, or where am I going wrong?
 A: 

Theorem:
Let $f$ be continuous on $[a, b]$ and differentiable on the open
interval $(a,b)$. Then
(a) $f$ is increasing in $[a,b]$ if $f′(x) > 0$ for each $x \in (a,  b)$
(b) $f$ is decreasing in $[a,b]$ if $f′(x) < 0$ for each $x \in (a,  b)$

I am thinking that the theorem is incomplete and the remarks are
correct. That theorem has to be as follows:
(a) $f$ is increasing in $[a,b]$ if $f′(x) \ge 0$ for each $x \in (a, 
> b)$
(b) $f$ is decreasing in $[a,b]$ if $f′(x) \le 0$ for each $x \in (a, 
> b)$

Indeed, the book's Theorem can be strengthened exactly as you suggest. (But it is not "incomplete"—just weaker than necessary/usual.)


Remarks:
(i) $f$ is strictly increasing in $(a, b)$ if $f′(x) > 0$ for each $x  \in (a, b)$
(ii) $f$ is strictly decreasing in $(a, b)$ if $f′(x) < 0$ for each $x  \in (a, b)$


In fact, the book's Remarks too can be strengthened:
(i) $f$ is strictly increasing in $[a, b]$ if $f′(x) > 0$ for each $x  \in (a, b)$
(ii) $f$ is strictly decreasing in $[a, b]$ if $f′(x) < 0$ for each $x  \in (a,b).$
