Difference between increasing function and strictly increasing function in terms of derivatives?

Question

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?

Answer 1

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. That theorem has to be as follows:

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

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

Indeed, the above theorem can be strengthened exactly as you suggest, and you can even scratch out its opening sentence's differentiability condition. (But the theorem isn't incomplete, just weaker than necessary.)

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 and expanded on:

(i) $f$ is strictly increasing at $x_0$ if $f′(x_0) > 0.$

(ii) $f$ is strictly decreasing at $x_0$ if $f′(x_0) < 0.$

(iii) $f′(x_0) \ge 0$ if $f$ is increasing at and differentiable at $x_0.$

(iii) $f′(x_0) \le 0$ if $f$ is decreasing at and differentiable at $x_0.$

---

ADDENDUM (with thanks to PierreCarre)

(c) $f$ is strictly increasing on $[a,b]$ if
    $f′(x) \ge 0$ for each $x \in (a, b)$ and $f'(x)=0$ only at isolated points.

(d) $f$ is strictly decreasing on $[a,b]$ if
    $f′(x) \le 0$ for each $x \in (a, b)$ and $f'(x)=0$ only at isolated points.