Monotonicity

Monotonicity

• $f$ is a real function that defined on interval $I$
  • $f$ is (weakly) increasing if $\forall x_1,x_2\in I,x_1<x_2⟹f(x_1)\le f(x_2)$
    • $f$ is (strictly) increasing if $\forall x_1,x_2\in I,x_1<x_2⟹f(x_1)<f(x_2)$
  • $f$ is (weakly) decreasing if $\forall x_1,x_2\in I,x_1<x_2⟹f(x_1)\ge f(x_2)$
    • $f$ is (strictly) decreasing if $\forall x_1,x_2\in I,x_1<x_2⟹f(x_1)>f(x_2)$
  • $f$ is monotonic if it’s increasing or decreasing

in this course monotonic is strictly monotonic

• (INFI2.d1.16) A function $f:[a,b]\to \mathbb{R}$ is said to be piecewise monotone if there exists a partition of $[a,b]$ into subintervals such that $f$ is weakly monotone on each internal of the subintervals
  • Each piecewise monotone function is also weakly monotone

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todo

• (5.40) if $f$ is increasing on neighborhood of $x_0$, then the one-side limits $\underset{x\to x_0^{-}}{\lim }f(x)$ and $\underset{x\to x_0^{+}}{\lim }f(x)$ exist and $\underset{x\to x_0^{-}}{\lim }f(x)\le f(x_0)\le \underset{x\to x_0^{+}}{\lim }f(x)$
• (5.41) Discontinuity of Monotonic Function is Jump Discontinuity
  • if $f$ is monotonic in $(a,b)$ and let $x_0\in (a.b)$ be a discontinuity point, then it’s the jump discontinuity point
• (5.42) $f$ is monotonic on $I$. if the interval image $f(I)$ is also an interval, then $f$ is continuous on $I$