Since, every function defined on closed and bounded interval is uniformly continuous function on [a, b],
$\therefore$ By definition of uniformly continuous function,
Suppose f is Riemann integrable on [a, b].
Let $\epsilon>0$ be arbitrary.
$\int\limits_\underline{a}^bf(x)dx=\int\limits_a^\underline{b}f(x)dx$Â Â Â ..... (1)
Let U & V be the two real vector spaces then a mapping $f:U\to V$ is said to be homomorphism (Linear Transformation) ifÂ
(i) $f(\alpha+\beta)=f(\alpha)+f(\beta), \forall \alpha, \beta\in U$Â
(ii) $f(c\alpha)=cf(\alpha), \forall c\in\mathbb{R}$ & $\alpha\in U$Â
Since, every function defined on closed and bounded interval is uniformly continuous function on [a, b],
$\therefore$ By definition of uniformly continuous function,
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