Defining Continuity at a Point

Continuity at a point x = c asks one question with three parts: does the graph pass through (c, f(c)) with no break, jump, or hole right there? To answer it you check f(c) exists, you check lim x→c f(x) exists (meaning the left and right limits agree), and you check that these two numbers match: lim x→c f(x) = f(c). Each condition fails in a different way. A function can have a limit but no defined value (a hole). It can have a defined value but no limit (a jump, where left and right sides head to different numbers). Or it can have both, but they disagree (a hole with a separate point plotted elsewhere). Only when all three line up is the function actually continuous there.

This is why one condition is never enough. A limit existing tells you the function is settling toward some number as you approach c, but says nothing about what's happening to f right at c, or whether f(c) is even defined. You need the handshake: value and limit must meet.

Checking continuity at a single point is the building block for checking it across an interval. A function is continuous on an interval when it passes this three-part test at every point in that interval, with one-sided versions of the test used at the endpoints. Later, when you classify discontinuities as removable, jump, or infinite, you're really just diagnosing which of the three conditions broke down at that one point.