One Delta for All

Real Analysis · Axiom Academy

Some functions need a different tolerance at every point. Others get away with one tolerance everywhere. That gap is the difference between ordinary continuity and uniform continuity. Continuity has a hidden catch: how fast it has to react A function is continuous at a point when you can hold the output inside any tolerance by keeping the input inside some tolerance . The quiet question nobody asks at first: does the same work everywhere, or does each point demand its own? For f(x) = x^2 the answer is brutal — drag right and the window you're allowed slams shut. Watch a fixed output tolerance ride along f(x) = x^2 . At each point, the green window is the exact input room you're allowed before f leaves the band. The curve gets steeper on the right, so the window keeps shrinking — there is no single that survives the whole trip. Same the whole way — but the it forces on you keeps getting smaller. The tolerance depends on where you are. Drive it yourself: pick a point, read off the it demands Drag the point along f(x) = x^2 and set the output tolerance . The readout is the exact largest input tolerance that keeps f inside the band: . Park the point near 0 , then drag it right with fixed — same band, steadily tighter . Hold steady and the demanded falls as x_0 grows — it never settles on a floor you could use for the whole line. That's the failure of uniform continuity. A function that hands you one and never asks again

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