mathematical-analysis-suppose-f-a-b-R-is-a-function-and-a-b-R-is-an-increasing-function-on-a-b-meanwhile-is-continuous-at-y-0-whe

Question Number 138733 by mnjuly1970 last updated on 17/Apr/21

\dots . . m a t h e m a t i c a l \dots . a n a l y s i s \dots . .

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) \dots .

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 \dots .