Question about increasing and decreasing functions

Question

Hi, I was studying the mean value theorem and increasing and decreasing functions. This question came to my mind. Can we say that $f$ increases on $[a,b]$?

Answer 1

To flesh out the comments while not aiming for maximum generality: If $X$ is a set of real numbers, and if $f$ is a real-valued function on $X$, we say $f$ is (strictly) increasing (on $X$) if $x_{1} < x_{2}$ implies $f(x_{1} < f(x_{2})$ for all $x_{1}$ and $x_{2}$ in $X$.

That is, "increasing functions preserve inequalities": Applying an increasing function to an inequality gives a new inequality of the same sign.

Notes:

• The term non-decreasing is similar, and means $x_{1} < x_{2}$ implies $f(x_{1} \leq f(x_{2})$. A strictly increasing function is non-decreasing. The term increasing may mean either non-decreasing or strictly increasing depending on the author. Authors often specify their convention early in the book. (Analogous remarks apply to the terms "decreasing", "non-increasing", and "strictly decreasing".)

• There is a well-known sufficient criterion in calculus for a function defined on an interval to be strictly increasing: Its derivative is positive everywhere. (Idea of proof: With the preceding notation, apply the mean value theorem to $f$ on $[x_{1}, x_{2}]$.)

• The well-known sufficient criterion is not necessary: $f(x) = x^{3}$ is strictly increasing, though $f'(0) = 0$. A strictly increasing differentiable function can have infinitely many critical points. In fact, the set of critical points can be an arbitrary closed set with empty interior. This point is often misunderstood.

• The well-known sufficient condition does not apply if the domain is not an interval. For example, $f(x) = -1/x$, defined on the set $X$ of non-zero real numbers (an interval with one point removed), has $f'(x) = 1/x^{2} < 0$ everywhere in its domain, but is not increasing: For example, $-1 < 1$, but $f(-1) = 1 > -1 = f(1)$.

• Monotone functions (which are non-decreasing or non-increasing) are among the nicest functions when we start digging into the theory underlying calculus. Particularly, a monotone function has one-sided limits at each interior point of its domain, whether or not the function is continuous there.