Defining Continuity

Calculus 1 · Axiom Academy

A curve you can draw without lifting your pencil — made precise by three conditions, and what breaks when even one fails. 1. Three Conditions, Checked at a Point Pick a point x = a on a curve. Continuity there is a single question asked three ways: is there a value f(a) , does the curve approach a single number as , and do those two agree ? Watch the limit close in from both sides and land exactly on the dot. f(a) is defined — the point exists on the graph (no gap, no division by zero). exists — the left and right approaches agree on one finite number. — the value the curve heads toward is the value that is actually there. 2. A Hole — When the Value Is Missing Take at x = 2 . Factor the top: x^2 - 4 = (x-2)(x+2) , so for every the function is just the line y = x + 2 . The curve sails toward height 4 from both sides — but at x = 2 the formula is , undefined. The limit exists; the value does not. 3. A Jump — When the Sides Disagree Now a function that steps. To the left of x = 1 it rides y = x + 1 ; from x = 1 on it rides y = x + 3 . Approaching from the left the curve heads for height 2 ; from the right, for height 4 . The two one-sided limits exist but disagree , so there is no single limit to speak of. 4. An Asymptote — When the Curve Runs Off to Infinity

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