Question

Q. If f, g is Reimann integrable on [a, b] then fg is reimann integrable on [a, b].

Answer 1

Answer on Question #68446 – Math – Real Analysis

Question

If $f, g$ is Riemann integrable on $[a, b]$ then $fg$ is Riemann integrable on $[a, b]$.

Solution

Function is Riemann integrable on $[a, b]$ if and only if set $S_f = \{x \in [a, b] \mid f \text{ is not continuous at } x\}$ of points of discontinuity of $f$ on $[a, b]$ has measure zero: $\mu(S_f) = 0$. Similarly, $\mu(S_g) = 0$.

If functions $f$ and $g$ are continuous at point $x$, then their product $fg$ is continuous at $x$ as well. Then $fg$ may be not continuous at $x$ when $f$ or $g$ is not continuous at $x$ (but not always, for example multiplying not continuous function by total zero gives total zero, which is continuous). Thus, the set of discontinuities of $fg$ is a subset of the sum of $S_f$ and $S_g: S_{fg} \subset S_f \cup S_g$. Measure is additive function, so $\mu(S_{fg}) = \mu(S_f \cup S_g) \leq \mu(S_f) + \mu(S_g) = 0 + 0 = 0$. Thus, function $fg$ is Riemann integrable on $[a, b]$.

Answer:

Yes, $fg$ is Riemann integrable on $[a,b]$.

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