Austin police reported the incident around 7:30 a.m. on South Congress Avenue, an area adjacent to a popular downtown shopping district known for its "I Love You So Much" mural.
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Officials said a man in his 20s was pronounced dead at the scene and a man in his 50s sustained life-threatening injuries. Two others were injured and treated at the scene.
Police said the incident started as an altercation at a coffee shop and escalated, leaving one person injured.
Patrons detained the attacker, but he broke away before authorities arrived. Officials said police chased the man on foot.
The man then allegedly stabbed two people at a nearby retail plaza, killing one, police said. The victims were found inside Freebirds World Burrito.
After the stabbing, the alleged attacker managed to get onto a building's rooftop and jump, police said.
Three people were taken to the hospital: the suspect, the stabbing victim and the coffee shop patron.
Stacy Romine said she witnessed the attack while out to get her morning coffee and then saw the attacker leap head-first from the roof of a nearby building.
"There was nothing that provoked it," Romine, 33, told The Associated Press. "This guy was not OK."
Romine said the man was "obviously under the influence of something" when he hit an elderly man at a coffee shop off a retail plaza.
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The identities of the suspect and victims were not released.
An investigation is ongoing.
The Associated Press contributed to this report.