lets-a-lt-b-and-f-a-b-R-integable-into-a-b-and-continuous-lets-I-a-closed-subset-of-a-b-proof-that-or-give-a-conter-example-I-fdx-0-I-a-b-f-0-

Question Number 1987 by 123456 last updated on 28/Oct/15

lets a < b and f : [a, b] \to R integable into [a, b] and continuous

lets I a closed subset of [a, b]

proof that (or give a conter example)

\int_{I} f d x = 0 \forall I \subset [a, b] \Rightarrow f = 0

Commented by prakash jain last updated on 28/Oct/15

If f is non - zero at some point c \in [a, b] then

f is continuous \Rightarrow

\exists ϵ > 0 such that \underset{c - ϵ}{\int^{c + ϵ}} f d x \neq 0