continuity_trick

continuity_trick - Passing the limit through a continuous...

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Passing the limit through a continuous function Math 8 2009, Chris Lyons Here’s a helpful trick that allows you to “pass” a limit from outside a continuous function to the inside. Proposition 1. Suppose that (i) g ( y ) is continuous at y = L , (ii) lim x a f ( x ) = L . Then lim x a g ( f ( x ) ) = g lim x a f ( x ) . Before giving a proof, here’s the intuitive idea behind the statement: let y = f ( x ). Then as x a we have y = f ( x ) L , and as y L we have g ( y ) g ( L ). In other words: as x a we have g ( f ( x ) ) g ( lim x a f ( x ) ) . Now here’s a real proof, using the theorem that the composition of two continuous functions is again continuous. Proof. Let’s make a new function: F ( x ) = f ( x ) if x = a L if x = a Then we know that F ( x ) is continuous at x = a because, by assumption (ii), we have lim x a F ( x ) = lim x a f ( x ) = L = F ( a ) . So now by Theorem 3.5 in the book, since g ( y ) is continuous at y = L and F ( x ) is continuous at x = a , the composition g ( F ( x )) is continuous at x = a . But what does this mean? It means lim x a g ( F ( x )) = g ( F ( a )) . Therefore lim x a g ( f ( x )) = lim x a g ( F ( x )) = g ( F ( a )) = g ( L ) = g lim x a f ( x ) . Here’s a handy consequence of this result: Corollary 2. Let g be a function that is continuous everywhere. If
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