In this video I go over a simple theorem on the continuity of composite function which when simply stated says that a continuous function of a continuous function is a continuous function. I also go over a few useful examples on the continuity of composite functions.

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Continuity of Composite Functions

Theorem

If g is continuous at a and f is continuous at g(a), then the composite function (f◦g)(x) = f(g(x)) is continuous at a.

“a continuous function of a continuous function is a continuous function”

Proof

Since g is continuous at a, we have:

Since f is continuous at g(a), we can apply the Limit of a Composite Function Theorem (see my earlier video) to obtain:

Where are the following functions continuous?