Types of Discontinuities

Real Analysis · Axiom Academy

LESSON Types of Discontinuities How a function can fail to be continuous at a point — removable, jump, and essential — and how to tell them apart with one-sided limits. Before classifying breaks, fix what an unbroken point looks like. f is continuous at c exactly when all three conditions hold — trace the graph and your pen never leaves the page. The mildest break. Both one-sided limits exist and are equal , so the two-sided limit exists — but f(c) is either undefined or sitting at the wrong value. The graph has a single hole you could patch by redefining f(c) to equal that common limit. at c = 1 . Factor: for , so both one-sided limits equal 2 — yet f(1) is , undefined. Defining f(1) = 2 removes the hole. Now both one-sided limits still exist and are finite — but they disagree . The graph leaps from one height to another, so the two-sided limit fails (condition 2). No single value of f(c) can close a gap, so this break is not removable. The size of the leap is exactly . and both exist and are finite, The floor function at c = 1 : as it stays at 0 , as it sits at 1 . Left limit 0 , right limit 1 — a finite jump of size 1 . (The Heaviside step H(x) jumps the same way at 0 .) 3. Essential (Infinite) Discontinuity The severe case: at least one one-sided limit fails to exist finitely . Either the function blows up without bound, or it oscillates forever without settling. No redefinition can fix a limit that simply isn't there. at least one one-sided limit is (unbounded), or

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