Riemann integral
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Definition
The Riemann integral of \(f\) equals \(S\) if the following condition holds:
For all \( \epsilon > 0 \), there exists a \( \delta \) such that for any tagged partition \( x_0, \ldots , x_n \) and \( t_0, \ldots , t_{n-1} \) whose mesh is less than \( \delta \), we have
\[ | \sum_{i=0}^{n-1} f(t_i) ( x_{i+1} - x_i) - S | < \epsilon \]
Properties
(Proof from first principle)
Reasons of failure
Evaluate the Riemann integral
\[ \int_0^1 \frac{1}{ \sqrt{x} } \, dx . \]
Note: The Riemann integral of \(f\) equals \(S\) if the following condition holds:
\( \forall \epsilon > 0 \), \( \exists \delta \) such that for any tagged partition \( x_0, \ldots , x_n \) and \( t_0, \ldots , t_{n-1} \) whose mesh is less than \( \delta \), we have
\[ \left| \sum_{i=0}^{n-1} f(t_i) ( x_{i+1} - x_i) - S \right| < \epsilon. \]
Inspiration.
Riemann-Lebesgue criterion
The function \( f: [ a , b ] \rightarrow \mathbb{R} \) is Riemann integrable if and only if (it is bounded and the set of points at which it is discontinuous has Lebesgue measure zero).