id	sid	eid	entity	type
ejpam-6496	1	1	european journal	ORG
ejpam-6496	1	2	2025	DATE
ejpam-6496	1	3	18	CARDINAL
ejpam-6496	1	4	3	CARDINAL
ejpam-6496	1	5	6496	CARDINAL
ejpam-6496	1	6	1307-5543	DATE
ejpam-6496	1	7	new york	GPE
ejpam-6496	1	8	riemann integrable	PERSON
ejpam-6496	1	9	prongjit1	PERSON
ejpam-6496	1	10	1	CARDINAL
ejpam-6496	1	11	kasetsart university	ORG
ejpam-6496	1	12	chalermphrakiat sakon	PERSON
ejpam-6496	1	13	sakon nakhon	PERSON
ejpam-6496	1	14	thailand	GPE
ejpam-6496	4	1	riemann integrable	PERSON
ejpam-6496	5	1	2020	DATE
ejpam-6496	5	2	26a24	CARDINAL
ejpam-6496	5	3	26a42	CARDINAL
ejpam-6496	5	4	riemann integrable	PERSON
ejpam-6496	5	5	1	CARDINAL
ejpam-6496	6	1	1	CARDINAL
ejpam-6496	6	2	1987	DATE
ejpam-6496	7	1	thomson	ORG
ejpam-6496	8	1	1989	DATE
ejpam-6496	8	2	2	CARDINAL
ejpam-6496	8	3	second	ORDINAL
ejpam-6496	9	1	3	CARDINAL
ejpam-6496	9	2	4	CARDINAL
ejpam-6496	11	1	1998	DATE
ejpam-6496	11	2	gordon [5	PERSON
ejpam-6496	13	1	6	CARDINAL
ejpam-6496	13	2	2014	DATE
ejpam-6496	13	3	2–4	CARDINAL
ejpam-6496	14	1	zheng	PERSON
ejpam-6496	14	2	shi	PERSON
ejpam-6496	15	1	8	CARDINAL
ejpam-6496	17	1	s. prongjit	PERSON
ejpam-6496	17	2	p. prasertsang	PERSON
ejpam-6496	18	1	1	CARDINAL
ejpam-6496	18	2	2025	CARDINAL
ejpam-6496	20	1	p. prasertsang	PERSON
ejpam-6496	21	1	j. pure	PERSON
ejpam-6496	21	2	18	CARDINAL
ejpam-6496	21	3	3	CARDINAL
ejpam-6496	21	4	2025	CARDINAL
ejpam-6496	21	5	6496 2	DATE
ejpam-6496	21	6	10	CARDINAL
ejpam-6496	21	7	6	CARDINAL
ejpam-6496	23	1	riemann integrability	PERSON
ejpam-6496	23	2	gordon	PERSON
ejpam-6496	23	3	5	CARDINAL
ejpam-6496	24	1	2	CARDINAL
ejpam-6496	24	2	∈	ORG
ejpam-6496	25	1	2.1	CARDINAL
ejpam-6496	26	1	1	CARDINAL
ejpam-6496	27	1	9	CARDINAL
ejpam-6496	28	1	1	CARDINAL
ejpam-6496	32	1	2	CARDINAL
ejpam-6496	33	1	9	CARDINAL
ejpam-6496	34	1	1	CARDINAL
ejpam-6496	36	1	3	CARDINAL
ejpam-6496	37	1	9	CARDINAL
ejpam-6496	38	1	1	CARDINAL
ejpam-6496	41	1	1	CARDINAL
ejpam-6496	42	1	1	CARDINAL
ejpam-6496	43	1	x0	PERSON
ejpam-6496	45	1	− δ(ti−1	PERSON
ejpam-6496	46	1	1	CARDINAL
ejpam-6496	46	2	s. prongjit	PERSON
ejpam-6496	46	3	p. prasertsang	PERSON
ejpam-6496	47	1	j. pure	PERSON
ejpam-6496	47	2	18	CARDINAL
ejpam-6496	47	3	3	CARDINAL
ejpam-6496	47	4	2025	CARDINAL
ejpam-6496	47	5	6496 3	DATE
ejpam-6496	49	1	1	CARDINAL
ejpam-6496	50	1	9	CARDINAL
ejpam-6496	51	1	2.2	CARDINAL
ejpam-6496	51	2	1	CARDINAL
ejpam-6496	52	1	9	CARDINAL
ejpam-6496	53	1	2	CARDINAL
ejpam-6496	54	1	9	CARDINAL
ejpam-6496	56	1	0	CARDINAL
ejpam-6496	56	2	4	CARDINAL
ejpam-6496	57	1	9	CARDINAL
ejpam-6496	58	1	1	CARDINAL
ejpam-6496	59	1	∈	ORG
ejpam-6496	59	2	r.	NORP
ejpam-6496	59	3	φ	ORG
ejpam-6496	61	1	1	CARDINAL
ejpam-6496	63	1	1	CARDINAL
ejpam-6496	63	2	0	CARDINAL
ejpam-6496	63	3	1	CARDINAL
ejpam-6496	64	1	φ	ORG
ejpam-6496	65	1	1	CARDINAL
ejpam-6496	65	2	4	CARDINAL
ejpam-6496	65	3	2	CARDINAL
ejpam-6496	65	4	5	CARDINAL
ejpam-6496	66	1	5	CARDINAL
ejpam-6496	66	2	riemann integrable	PERSON
ejpam-6496	68	1	1	CARDINAL
ejpam-6496	70	1	i=1 ω(f	PERSON
ejpam-6496	70	2	ω(f	PERSON
ejpam-6496	75	1	3	CARDINAL
ejpam-6496	75	2	3.1	CARDINAL
ejpam-6496	75	3	3	CARDINAL
ejpam-6496	75	4	4	CARDINAL
ejpam-6496	75	5	3	CARDINAL
ejpam-6496	76	1	4	CARDINAL
ejpam-6496	77	1	1	CARDINAL
ejpam-6496	77	2	3	CARDINAL
ejpam-6496	78	1	s. prongjit	PERSON
ejpam-6496	78	2	p. prasertsang	PERSON
ejpam-6496	79	1	j. pure	PERSON
ejpam-6496	79	2	18	CARDINAL
ejpam-6496	79	3	3	CARDINAL
ejpam-6496	79	4	2025	CARDINAL
ejpam-6496	79	5	6496 4	DATE
ejpam-6496	79	6	10 x	QUANTITY
ejpam-6496	79	7	1	CARDINAL
ejpam-6496	79	8	c2 c3 c4 j1	PERSON
ejpam-6496	79	9	j2 j3 j4 j5	ORG
ejpam-6496	79	10	2	CARDINAL
ejpam-6496	79	11	5∑	CARDINAL
ejpam-6496	79	12	1	CARDINAL
ejpam-6496	79	13	4	CARDINAL
ejpam-6496	79	14	2	CARDINAL
ejpam-6496	81	1	0	CARDINAL
ejpam-6496	81	2	0	CARDINAL
ejpam-6496	83	1	0	CARDINAL
ejpam-6496	85	1	0	CARDINAL
ejpam-6496	85	2	0	CARDINAL
ejpam-6496	86	1	x−	PERSON
ejpam-6496	86	2	⊆	CARDINAL
ejpam-6496	86	3	f(y)− f(x	ORG
ejpam-6496	89	1	three	CARDINAL
ejpam-6496	90	1	1	CARDINAL
ejpam-6496	90	2	f(s	GPE
ejpam-6496	91	1	2	CARDINAL
ejpam-6496	91	2	f(s	GPE
ejpam-6496	92	1	1	CARDINAL
ejpam-6496	93	1	2	CARDINAL
ejpam-6496	93	2	0	CARDINAL
ejpam-6496	94	1	f(s	GPE
ejpam-6496	95	1	3	CARDINAL
ejpam-6496	95	2	2	CARDINAL
ejpam-6496	98	1	3	CARDINAL
ejpam-6496	102	1	lemma 2	ORG
ejpam-6496	102	2	0	CARDINAL
ejpam-6496	103	1	δ(x	PERSON
ejpam-6496	104	1	1	CARDINAL
ejpam-6496	106	1	s. prongjit	PERSON
ejpam-6496	106	2	p. prasertsang	PERSON
ejpam-6496	107	1	j. pure	PERSON
ejpam-6496	107	2	18	CARDINAL
ejpam-6496	107	3	3	CARDINAL
ejpam-6496	107	4	2025	CARDINAL
ejpam-6496	107	5	6496 5	DATE
ejpam-6496	107	6	10	CARDINAL
ejpam-6496	107	7	two	CARDINAL
ejpam-6496	108	1	1	CARDINAL
ejpam-6496	108	2	x0	ORG
ejpam-6496	111	1	x0	ORG
ejpam-6496	115	1	2	CARDINAL
ejpam-6496	115	2	x0	ORG
ejpam-6496	117	1	two	CARDINAL
ejpam-6496	121	1	1	CARDINAL
ejpam-6496	124	1	3	CARDINAL
ejpam-6496	124	2	two	CARDINAL
ejpam-6496	125	1	1	CARDINAL
ejpam-6496	129	1	f(y)− f(x	ORG
ejpam-6496	130	1	0	CARDINAL
ejpam-6496	130	2	2	CARDINAL
ejpam-6496	132	1	4	CARDINAL
ejpam-6496	135	1	3	CARDINAL
ejpam-6496	135	2	1	CARDINAL
ejpam-6496	136	1	d.	NORP
ejpam-6496	138	1	f(b	GPE
ejpam-6496	138	2	f(s	GPE
ejpam-6496	139	1	f(b	GPE
ejpam-6496	141	1	2	CARDINAL
ejpam-6496	141	2	1	CARDINAL
ejpam-6496	141	3	f s. prongjit	PERSON
ejpam-6496	141	4	p. prasertsang	PERSON
ejpam-6496	142	1	j. pure	PERSON
ejpam-6496	142	2	18	CARDINAL
ejpam-6496	142	3	3	CARDINAL
ejpam-6496	142	4	2025	CARDINAL
ejpam-6496	142	5	6496 6	DATE
ejpam-6496	143	1	4	CARDINAL
ejpam-6496	143	2	two	CARDINAL
ejpam-6496	143	3	9	CARDINAL
ejpam-6496	144	1	5	CARDINAL
ejpam-6496	151	1	4	CARDINAL
ejpam-6496	153	1	g(b	GPE
ejpam-6496	155	1	0	CARDINAL
ejpam-6496	158	1	6	CARDINAL
ejpam-6496	160	1	f(b)− f(a	ORG
ejpam-6496	163	1	4	CARDINAL
ejpam-6496	166	1	5	CARDINAL
ejpam-6496	166	2	h(a	ORG
ejpam-6496	167	1	0	CARDINAL
ejpam-6496	168	1	g′(c)−	ORG
ejpam-6496	171	1	s. prongjit	PERSON
ejpam-6496	171	2	p. prasertsang	PERSON
ejpam-6496	172	1	j. pure	PERSON
ejpam-6496	172	2	18	CARDINAL
ejpam-6496	172	3	3	CARDINAL
ejpam-6496	172	4	2025	CARDINAL
ejpam-6496	172	5	6496 7	DATE
ejpam-6496	173	1	riemann integral via	ORG
ejpam-6496	173	2	riemann integrable	PERSON
ejpam-6496	174	1	5	CARDINAL
ejpam-6496	174	2	riemann	PERSON
ejpam-6496	174	3	gordon	PERSON
ejpam-6496	175	1	5	CARDINAL
ejpam-6496	176	1	7	CARDINAL
ejpam-6496	177	1	riemann integrable	PERSON
ejpam-6496	179	1	0	CARDINAL
ejpam-6496	180	1	δ(x	PERSON
ejpam-6496	182	1	1	CARDINAL
ejpam-6496	184	1	1	CARDINAL
ejpam-6496	185	1	ω(f	PERSON
ejpam-6496	187	1	2ε	CARDINAL
ejpam-6496	188	1	1	CARDINAL
ejpam-6496	190	1	i=1 ω(f	PERSON
ejpam-6496	191	1	f(xi−1))2ε	ORG
ejpam-6496	191	2	2ε(f(b)−	CARDINAL
ejpam-6496	192	1	riemann integrable	PERSON
ejpam-6496	194	1	3	CARDINAL
ejpam-6496	197	1	1	CARDINAL
ejpam-6496	199	1	2l	CARDINAL
ejpam-6496	200	1	4	CARDINAL
ejpam-6496	204	1	1	CARDINAL
ejpam-6496	206	1	at least one	CARDINAL
ejpam-6496	206	2	∈	ORG
ejpam-6496	206	3	0	CARDINAL
ejpam-6496	206	4	1	CARDINAL
ejpam-6496	208	1	two	CARDINAL
ejpam-6496	209	1	1	CARDINAL
ejpam-6496	210	1	x0	ORG
ejpam-6496	211	1	xn−1	GPE
ejpam-6496	212	1	two	CARDINAL
ejpam-6496	213	1	2	CARDINAL
ejpam-6496	213	2	1	CARDINAL
ejpam-6496	215	1	l − 1	DATE
ejpam-6496	216	1	two	CARDINAL
ejpam-6496	216	2	two	CARDINAL
ejpam-6496	217	1	2l	CARDINAL
ejpam-6496	218	1	8	CARDINAL
ejpam-6496	219	1	riemann integrable	PERSON
ejpam-6496	220	1	s. prongjit	PERSON
ejpam-6496	220	2	p. prasertsang	PERSON
ejpam-6496	221	1	j. pure	PERSON
ejpam-6496	221	2	18	CARDINAL
ejpam-6496	221	3	3	CARDINAL
ejpam-6496	221	4	2025	CARDINAL
ejpam-6496	221	5	6496 8	DATE
ejpam-6496	223	1	δ(x	PERSON
ejpam-6496	224	1	k = max{|kj1 −kj2	PERSON
ejpam-6496	224	2	j1	PERSON
ejpam-6496	224	3	j2 = 1	PERSON
ejpam-6496	226	1	1	CARDINAL
ejpam-6496	226	2	2	DATE
ejpam-6496	228	1	1	CARDINAL
ejpam-6496	229	1	two	CARDINAL
ejpam-6496	230	1	1	CARDINAL
ejpam-6496	231	1	⊆	CARDINAL
ejpam-6496	231	2	1	CARDINAL
ejpam-6496	231	3	2	DATE
ejpam-6496	234	1	0	CARDINAL
ejpam-6496	234	2	2	CARDINAL
ejpam-6496	234	3	1	CARDINAL
ejpam-6496	234	4	2	DATE
ejpam-6496	240	1	2ε	CARDINAL
ejpam-6496	240	2	1	CARDINAL
ejpam-6496	242	1	1	CARDINAL
ejpam-6496	242	2	2	DATE
ejpam-6496	244	1	3	CARDINAL
ejpam-6496	247	1	ω(f	PERSON
ejpam-6496	247	2	2ε	CARDINAL
ejpam-6496	248	1	4lε	ORDINAL
ejpam-6496	249	1	riemann integrable	PERSON
ejpam-6496	250	1	4	CARDINAL
ejpam-6496	252	1	1	CARDINAL
ejpam-6496	255	1	ω(f	PERSON
ejpam-6496	255	2	1	CARDINAL
ejpam-6496	255	3	j = 1	PERSON
ejpam-6496	259	1	1	CARDINAL
ejpam-6496	261	1	1	CARDINAL
ejpam-6496	261	2	1 m m∑	QUANTITY
ejpam-6496	261	3	ω(f	PERSON
ejpam-6496	262	1	ω(f	PERSON
ejpam-6496	262	2	ω(f	PERSON
ejpam-6496	263	1	ω(f	PERSON
ejpam-6496	263	2	1	CARDINAL
ejpam-6496	264	1	ω(f	PERSON
ejpam-6496	264	2	1	CARDINAL
ejpam-6496	265	1	1	CARDINAL
ejpam-6496	266	1	∈	ORG
ejpam-6496	266	2	0 ≤ ωj	QUANTITY
ejpam-6496	266	3	1	CARDINAL
ejpam-6496	267	1	s. prongjit	PERSON
ejpam-6496	267	2	p. prasertsang	PERSON
ejpam-6496	268	1	j. pure	PERSON
ejpam-6496	268	2	18	CARDINAL
ejpam-6496	268	3	3	CARDINAL
ejpam-6496	268	4	2025	CARDINAL
ejpam-6496	268	5	6496 9	DATE
ejpam-6496	268	6	10	CARDINAL
ejpam-6496	268	7	1	CARDINAL
ejpam-6496	269	1	4	CARDINAL
ejpam-6496	269	2	ω(f	PERSON
ejpam-6496	270	1	1	CARDINAL
ejpam-6496	270	2	9	CARDINAL
ejpam-6496	271	1	riemann integrable	PERSON
ejpam-6496	273	1	1	CARDINAL
ejpam-6496	275	1	ω(f	PERSON
ejpam-6496	275	2	1	CARDINAL
ejpam-6496	276	1	lemma 4	ORG
ejpam-6496	276	2	1	CARDINAL
ejpam-6496	278	1	ω(f	PERSON
ejpam-6496	280	1	0	CARDINAL
ejpam-6496	281	1	δ(x	PERSON
ejpam-6496	282	1	1	CARDINAL
ejpam-6496	282	2	2	DATE
ejpam-6496	284	1	1	CARDINAL
ejpam-6496	286	1	1	CARDINAL
ejpam-6496	286	2	2	DATE
ejpam-6496	289	1	m̂	GPE
ejpam-6496	291	1	j = 2	LAW
ejpam-6496	292	1	[ yj−1, yj	ORG
ejpam-6496	293	1	1	CARDINAL
ejpam-6496	295	1	4.	CARDINAL
ejpam-6496	296	1	lemma 2.	ORG
ejpam-6496	296	2	3	CARDINAL
ejpam-6496	296	3	6	CARDINAL
ejpam-6496	297	1	4	CARDINAL
ejpam-6496	297	2	two	CARDINAL
ejpam-6496	298	1	riemann integrable function	PERSON
ejpam-6496	298	2	three	CARDINAL
ejpam-6496	299	1	gordon	PERSON
ejpam-6496	299	2	riemann integrability	PERSON
ejpam-6496	299	3	three	CARDINAL
ejpam-6496	302	1	s. prongjit	PERSON
ejpam-6496	302	2	p. prasertsang	PERSON
ejpam-6496	303	1	j. pure	PERSON
ejpam-6496	303	2	18	CARDINAL
ejpam-6496	303	3	3	CARDINAL
ejpam-6496	303	4	2025	CARDINAL
ejpam-6496	303	5	6496 10	DATE
ejpam-6496	303	6	10	CARDINAL
ejpam-6496	303	7	kasetsart university	ORG
ejpam-6496	303	8	chalermprakiat sakon nakhon	PERSON
ejpam-6496	303	9	sakon nakhon	PERSON
ejpam-6496	303	10	thailand	GPE
ejpam-6496	304	1	european journal	ORG
ejpam-6496	305	1	1	CARDINAL
ejpam-6496	305	2	m. w. botsko	PERSON
ejpam-6496	307	1	american	NORP
ejpam-6496	307	2	monthly	DATE
ejpam-6496	307	3	94:450–452	CARDINAL
ejpam-6496	307	4	1987	DATE
ejpam-6496	308	1	2	CARDINAL
ejpam-6496	308	2	m. w. botsko	PERSON
ejpam-6496	310	1	american	NORP
ejpam-6496	310	2	monthly	DATE
ejpam-6496	310	3	96:328–333	CARDINAL
ejpam-6496	310	4	1989	DATE
ejpam-6496	311	1	3	CARDINAL
ejpam-6496	311	2	k. klaimon	PERSON
ejpam-6496	313	1	9:156–161	CARDINAL
ejpam-6496	313	2	1990	DATE
ejpam-6496	314	1	4	CARDINAL
ejpam-6496	314	2	k. zangara	PERSON
ejpam-6496	314	3	j. marafino	PERSON
ejpam-6496	316	1	2009	DATE
ejpam-6496	317	1	5	CARDINAL
ejpam-6496	317	2	r. a. gordon	PERSON
ejpam-6496	319	1	american	NORP
ejpam-6496	319	2	monthly	DATE
ejpam-6496	319	3	105(1):107–117, 1998	DATE
ejpam-6496	320	1	6	CARDINAL
ejpam-6496	320	2	w. sodsiri	PERSON
ejpam-6496	322	1	far east journal of mathematical sciences	ORG
ejpam-6496	322	2	91(1):97–109, 2014	DATE
ejpam-6496	323	1	7	CARDINAL
ejpam-6496	323	2	w. sodsiri	PERSON
ejpam-6496	325	1	far east journal of mathematical sciences	ORG
ejpam-6496	325	2	91(2):233–245	ORG
ejpam-6496	325	3	2014	DATE
ejpam-6496	326	1	8	CARDINAL
ejpam-6496	326	2	zheng	PERSON
ejpam-6496	326	3	x. shi	PERSON
ejpam-6496	328	1	329:344–352,	DATE
ejpam-6496	328	2	2018	DATE
ejpam-6496	329	1	9	CARDINAL
ejpam-6496	329	2	r. g. bartle	PERSON
ejpam-6496	329	3	d. r. sherbert	PERSON
ejpam-6496	331	1	john wiley & sons	ORG
ejpam-6496	331	2	new york	GPE
ejpam-6496	331	3	2011	DATE