Question Number 138733 by mnjuly1970 last updated on 17/Apr/21
![.....mathematical ....analysis..... suppose f :[a , b]→R is a function and α:[a , b]→^(α↗) R (α is an increasing function on [a , b]) meanwhile α is continuous at y_0 where a≤y_0 ≤b . defining f(x)= { (( 1 x=y_0 )),(( 0 x≠y_0 )) :} prove that : f∈ R (α) .... Hint: f∈R (α) ⇔ ∀ ε>0 ∃ P_ε ; U(P_ε ,f,α)−L(P_ε ,f,α)<ε Reimann criterion ....](Q138733.png)
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:....{analysis}..... \\ $$$$\:\:{suppose}\:\:\:\:{f}\::\left[{a}\:,\:{b}\right]\rightarrow\mathbb{R}\:{is}\:{a}\:{function} \\ $$$$\:\:\:{and}\:\:\:\alpha:\left[{a}\:,\:{b}\right]\overset{\alpha\nearrow} {\rightarrow}\mathbb{R}\:\left(\alpha\:{is}\:{an}\:{increasing}\:{function}\right. \\ $$$$\left.\:{on}\:\left[{a}\:,\:{b}\right]\right)\:\:{meanwhile}\:\alpha\:{is}\:{continuous}\:{at}\:{y}_{\mathrm{0}} \: \\ $$$$\:\:{where}\:\:\:{a}\leqslant{y}_{\mathrm{0}} \leqslant{b}\:\:.\:{defining}\: \\ $$$$\:\:\:{f}\left({x}\right)=\begin{cases}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{x}={y}_{\mathrm{0}} }\\{\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:{x}\neq{y}_{\mathrm{0}} }\end{cases} \\ $$$$\:\:\:\:{prove}\:\:{that}\::\:{f}\in\:\mathscr{R}\:\left(\alpha\right)\:.... \\ $$$$\:\:\:\:{Hint}:\:{f}\in\mathscr{R}\:\left(\alpha\right)\:\Leftrightarrow\:\forall\:\epsilon>\mathrm{0}\:\exists\:{P}_{\epsilon} \:;\:{U}\left({P}_{\epsilon} ,{f},\alpha\right)−{L}\left({P}_{\epsilon} ,{f},\alpha\right)<\epsilon \\ $$$$\:\:\:\:{Reimann}\:\:{criterion}\:.... \\ $$