Algebra of Continuous Functions

Real Analysis · Axiom Academy

LESSON Algebra of Continuous Functions Build complex continuous functions from simple ones — how sums, products, quotients, and compositions preserve continuity. 1. Sum of Continuous Functions If neither function has a jump or break, adding them point-by-point can't create one either — at every x you just stack one height on the other, and the combined curve stays unbroken. 2. Product of Continuous Functions Multiplying two continuous functions also gives a continuous one. Here a vertical strip whose height is the product sweeps across — and the product curve it traces never tears. Scaling a continuous function vertically — stretching it by a factor k — never tears it. Watch the same curve grow as k climbs from 1 to 2 : it stretches, but stays in one piece. (This is just the product rule with g the constant function k .) 4. Quotient of Continuous Functions Division preserves continuity, with one caveat: the denominator must be nonzero. Here never drops below 1 (the dashed floor) — so g is never 0 and the quotient sweeps out a curve with no breaks. 5. Composition of Continuous Functions Composition chains continuity. Send an input x through g , then push the result through f : a point on the input axis maps to g(x) , which maps to f(g(x)) . As x slides, the whole chain moves together — no link ever snaps. 6. Polynomials, Rationals, and Where They Break These rules give a toolkit for recognizing continuous functions instead of checking limits one by one:

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