Varanasi is routinely called the oldest continuously inhabited city on earth. The claim is worth taking apart, because what supports it is not what most people assume.
What the ground says
Excavation at Rajghat inside the city has produced material from around 800 BCE, and the nearby site of Aktha has yielded earlier dates still. That is genuinely ancient, contemporary with the early Iron Age in the Ganges plain, and older than Rome. It is not, however, evidence of continuity. Continuous habitation is a much harder thing to demonstrate than early habitation, and the archaeological sequence at Varanasi has gaps in it like any other.
The competing candidates (Jericho, Byblos, Damascus, Aleppo), are each argued for on similarly incomplete grounds. The honest position is that Varanasi is among a small group of cities with a very long and probably unbroken record, and that ranking them precisely is not something the evidence supports.
What the texts say
The city appears in the Rigveda as Kashi, and in Buddhist and Jain sources as a major centre well before the common era, the Buddha's first sermon was preached a short walk away at Sarnath, and he is described as going to Varanasi because that is where the people were. The Chinese pilgrim Xuanzang, passing through around 635, described a large and prosperous religious city with thousands of monks and dozens of temples.
What is actually old, and what is not
Very little standing in Varanasi is ancient. The ghats that define the city are overwhelmingly eighteenth-century work, built or rebuilt under Maratha patronage, Dashashwamedh reached its present form under the Peshwas in the 1740s and was rebuilt by Ahilyabai Holkar of Indore in 1774. The Kashi Vishwanath temple dates from 1780. The repeated destructions of the medieval period are the reason: what a pilgrim sees is a city rebuilt several times over on a very old site.
The Ganga Aarti at Dashashwamedh, now the single most photographed thing in the city and widely assumed to be immemorial, is a modern devotional performance in its present form.
None of this diminishes the claim the city makes for itself. It relocates it. What has continued at Varanasi is not a set of buildings but a practice (bathing, dying, burning), and the practice is demonstrably far older than anything holding it up.