Proving f(x) = x² is Continuous at x = 2

Real Analysis · Axiom Academy

EXAMPLE Proving f(x)=x^2 is Continuous at x=2 An – proof: working backwards to choose . Using the – definition of continuity, prove that f(x)=x^2 is continuous at x=2 — that is, show . Goal of the proof: shrink the red -interval around x=2 until the curve carries it inside the green -band around y=4 . (Static figure — drawn once.) Nice work — you built a complete – proof of continuity. Here is what made it go through: The definition is precise: for every we must produce a with . Work backwards: start from and reverse-engineer what has to be. Factoring is the key move: isolates the factor |x-2| that controls. Bound the leftover factor: pre-restricting forces , hence . The minimum trick: honours both the restriction and the requirement at once. The same "factor, bound, then take a minimum" recipe works for every polynomial — which is exactly why polynomials are continuous everywhere.

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