Defining Continuity at a Point

Drawing a graph without lifting your pencil feels simple, but to pin down exactly when a graph has no break at a point, you need three separate checks. First, f(c) must exist — the function actually has a value at that point, so there's a dot on the graph, not a hole. Second, the limit as x approaches c must exist — the graph approaches the same height from the left and from the right. Third, that limit must equal f(c) — the height the graph is heading toward is the same height where the dot actually sits. Each condition rules out a different kind of break: no f(c) means a hole with nothing filling it, no limit means a jump where the two sides disagree, and a limit that doesn't match f(c) means a hole with the dot relocated somewhere else.

This definition is the rigorous version of the pencil test, and it's what makes continuity checkable algebraically instead of just visually. For piecewise functions, you usually only worry about the seam point, since each piece is continuous on its own by construction. You check continuity there by plugging the boundary value into both pieces and setting them equal, which often means solving for an unknown constant.

This topic sets up everything that follows in the unit: the Intermediate Value Theorem, removable versus non-removable discontinuities, and later the idea that differentiability requires continuity first. Get comfortable testing all three conditions here, and those later ideas will just be refinements of this same checklist.