.....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


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Question Number 138733 by mnjuly1970 last updated on 17/Apr/21
$.....mathematical ....analysis..... suppose f :[a , b]→R is a function and α:[a , b]→^(α↗) R (α is an increasing function on [a , b]) meanwhile α is continuous at y_0 where a≤y_0 ≤b . defining f(x)= { (( 1 x=y_0 )),(( 0 x≠y_0 )) :} prove that : f∈ R (α) .... Hint: f∈R (α) ⇔ ∀ ε>0 ∃ P_ε ; U(P_ε ,f,α)−L(P_ε ,f,α)<ε Reimann criterion ....$
$. . . . . m a t h e m a t i c a l . . . . a n a l y s i s . . . . .$ $s u p p o s e f : [a, b] \to R i s a f u n c t i o n$ $a n d α : [a, b] \overset{α ↗}{\to} R (α i s a n i n c r e a s i n g f u n c t i o n$ $o n [a, b]) m e a n w h i l e α i s c o n t i n u o u s a t y_{0}$ $w h e r e a ⩽ y_{0} ⩽ b . d e f i n i n g$ $f (x) = {\begin{cases} 1 x = y_{0} \\ 0 x \neq y_{0} \end{cases}$ $p r o v e t h a t : f \in R (α) . . . .$ $H i n t : f \in R (α) \Leftrightarrow \forall ϵ > 0 \exists P_{ϵ}; U (P_{ϵ}, f, α) - L (P_{ϵ}, f, α) < ϵ$ $R e i m a n n c r i t e r i o n . . . .$
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