\displaystyle 1.\ \text{A function } f(x) \text{ is said to be a strictly increasing function on } (a,b) \\ \text{ if } x_1<x_2 \Rightarrow f(x_1)<f(x_2)\ \text{for all } x_1,x_2\in(a,b).
\displaystyle \text{If } x_1<x_2 \Rightarrow f(x_1)>f(x_2)\ \text{for all } x_1,x_2\in(a,b),\ \text{then } f(x) \text{ is said to be strictly} \\ \text{decreasing on } (a,b).

\displaystyle 2.\ \text{A function } f(x) \text{ is said to be monotonic on } (a,b) \text{ if it is either strictly increasing or} \\ \text{strictly decreasing on } (a,b).

\displaystyle 3.\ \text{A function } f(x) \text{ is said to be increasing (decreasing) at a point } x_0,\ \text{if there is an} \\ \text{interval } (x_0-h,x_0+h) \text{ containing } x_0 \text{ such that } f(x) \text{ is increasing (decreasing) on } (x_0-h,x_0+h).

\displaystyle 4.\ \text{A function } f(x) \text{ is said to be increasing (decreasing) on } [a,b],\ \text{if it is increasing} \\ \text{(decreasing) on } (a,b) \text{ and it is increasing (decreasing) at } x=a \text{ and } x=b.

\displaystyle 5.\ \text{The necessary and sufficient condition for a differentiable function defined on } (a,b) \text{ to be} \\ \text{strictly increasing on } (a,b) \text{ is that } f'(x)>0 \text{ for all } x\in(a,b).

\displaystyle 6.\ \text{The necessary and sufficient condition for a differentiable function defined on } (a,b) \text{ to be} \\ \text{strictly decreasing on } (a,b) \text{ is that } f'(x)<0 \text{ for all } x\in(a,b).

\displaystyle 7.\ \text{Let } f(x) \text{ be a function defined on } (a,b).
\displaystyle (a)\ \text{If } f'(x)>0 \text{ for all } x\in(a,b) \text{ except for a finite number of points, where } \\ f'(x)=0,\ \text{then } f(x) \text{ is increasing on } (a,b).
\displaystyle (b)\ \text{If } f'(x)<0 \text{ for all } x\in(a,b) \text{ except for a finite number of points, where } \\ f'(x)=0,\ \text{then } f(x) \text{ is decreasing on } (a,b).

\displaystyle 8.\ (i)\ \text{If } f(x) \text{ is strictly increasing function on an interval } [a,b],\ \text{then } f^{-1} \text{ exists and it is also a} \\ \text{strictly increasing function.}

\displaystyle (ii)\ \text{If } f(x) \text{ is strictly increasing function on an interval } [a,b] \text{ such that it is continuous,} \\ \text{then } f^{-1} \text{ is continuous on } [f(a),f(b)].

\displaystyle (iii)\ \text{If } f(x) \text{ is continuous on } [a,b] \text{ such that } f'(c)\ge 0\ (f'(c)>0) \text{ for each } c\in(a,b),\ \text{then } f(x) \text{ is monotonically (strictly) increasing function on } [a,b].

\displaystyle (iv)\ \text{If } f(x) \text{ is continuous on } [a,b] \text{ such that } f'(c)\le 0\ (f'(c)<0) \text{ for each } c\in(a,b),\ \text{then } f(x) \text{ is monotonically (strictly) decreasing function on } [a,b].

\displaystyle (v)\ \text{If } f(x) \text{ and } g(x) \text{ are monotonically (or strictly) increasing (or decreasing) functions} \\ \text{on } [a,b],\ \text{then } g\circ f(x) \text{ is a monotonically (or strictly) increasing function on } [a,b].

\displaystyle (vi)\ \text{If one of the two functions } f(x) \text{ and } g(x) \text{ is strictly (or monotonically) increasing} \\ \text{and other a strictly (monotonically) decreasing, then } g\circ f(x) \text{ is strictly (monotonically)} \\ \text{decreasing on } [a,b].

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