Let f:[a, b] →ℝ be a function. Show that f is integrable if...

Let $f:[a, b] \rightarrow \mathbb{R}$ be a function. Show that $f$ is integrable if (i) $r f$ is integrable for some nonzero $r \in \mathbb{R}$, or (ii) if $f$ is bounded, $f(x) \neq 0$ for all $x \in[a, b]$, and $1 / f$ is integrable.

Key Concepts

Riemann Integrability

This concept refers to the criteria for a function to be Riemann integrable, typically characterized by the equality of the upper and lower sums or by the condition that the set of discontinuities of the function has measure zero. It is the foundational definition used to determine whether the integral of a function over an interval is well-defined.

Linearity of the Integral

Linearity deals with the property that the integral of a constant multiple of a function equals the constant multiplied by the integral of that function. This property is crucial in showing that if a nonzero scalar multiple of a function is integrable, then the function itself is integrable, since the operation can be reversed by dividing by the nonzero constant.

Boundedness and Nonzero-ness

A function being bounded ensures that its values do not diverge, which is a necessary condition for integrability. The additional requirement that the function does not vanish on the interval guarantees that its reciprocal is well-defined and avoids issues such as division by zero. These conditions are instrumental in arguments related to the integrability of reciprocal functions.

Reciprocal Functions in Integration

This concept examines the relationship between a function and its reciprocal, particularly under the conditions where the reciprocal is integrable. When a bounded function is nonzero and its reciprocal is integrable, this relationship can often be used to deduce properties about the integrability of the original function, leveraging the interplay between the two functions.

Transcript

Please give Ace some feedback