If Riemann integrable, then it has finite number of discontinuities
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Example of Riemann integrable $f: [0,1] \to \mathbb R $ whose set of discontinuity points is an uncountable and dense set in $[0,1]$
58087
Riemann rearrangement theorem
115878
Continuous unbounded but integrable functions
6990
For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
152563
Set of discontinuity of monotone function is countable
86618
Approximating Riemann integrable functions
99485
Prove that $1/f$ is Riemann integrable on $[a,b]$
14117
Holomorphic function has at most countably zeros
4090
Show that $f$ is integrable iff $\sum\limits_{n\in\mathbb{Z}}2^n \mu(\{x\in S:2^n< \left|f(x)\right|\leq2^{n+1}\})<\infty.$
116622
What's the "limit" in the definition of Riemann integrals?
11167
A counterexample of a differentiable function on an interval whose derivative is discontinuous
18942
An equivalent definition of uniform integrability
22210
Points of discontinuity of a bijective function $f:\mathbb{R} \to [0,\infty)$
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Real Analysis Riemann Integration - Strict Monotonicity for Integrals
18222
Set of continuity points of a real function
112111
Riemann Intergral proof question
71095
Is Dirichlet function Riemann integrable?