Continuity for Multivariable Functions

Calculus 3 · Axiom Academy

LESSON Continuity for Multivariable Functions f is continuous at (a,b) exactly when every path of approach agrees — and agrees with f(a,b) . Continuity at a point is three separate demands. Miss any one of them and the function is discontinuous there. f(a,b) is defined — the point lies in the domain. The limit exists : for one single number L , the same along every path. On the real line a limit only has to survive two approaches, from the left and from the right. In the plane a point can be reached along infinitely many paths, and the limit must come out the same on all of them — a far stronger requirement. Every path into (a,b) delivers the same limit L And that limit is the value the function already has there Checking three conditions from scratch for every function would be exhausting. The composition theorem does the work for you: if g(x,y) is continuous at (a,b) and the single-variable function f(u) is continuous at u = g(a,b) , then the composite is continuous at (a,b) . Polynomials in x and y , together with , and , are continuous everywhere. Rational functions and are continuous on their domains. e^ x^2+y^2 and are continuous on all of . is continuous everywhere except the origin, where its input hits 0 and leaves the domain of . e^ x^2+y^2 is with g(x,y) = x^2+y^2 and f(u) = e^ u . At the inner map gives u = 0.36 + 0.64 = 1 , and the outer map gives . Neither stage jumps, so the composite cannot jump either.

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