This popular Algorithms book tells us the following definition of monotonically increasing functions:
A function $f(n)$ is monotonically increasing if $m \le n$ implies $f(m) \le f(n)$.
I can see why we need $m < n \implies f(m) \le f(n)$ but I am having a hard time understanding why we need to write $m \le n$ instead of just $m < n$. Isn't it kind of redundant because $m = n \implies f(m) = f(n)$ is already true because that's what makes $f(n)$ a "function" in the first place?
So re-writing the above definition:
A function $f(n)$ is monotonically increasing if $m < n$ implies $f(m) \le f(n)$.
Is this not sufficient? Or am I missing something?