Composites of continuous functions Figure A.21 gives the diagram for a proof that the composite of two continuous functions is continuous. Reconstruct the proof from the diagram. The statement to be proved is this: If ƒ is continuous at x = c and g is continuous at ƒ(c), then is continuous at c.
Assume that c is an interior point of the domain of ƒ and that ƒ(c) is an interior point of the domain of g. This will make the limits involved two-sided. (The arguments for the cases that involve one-sided limits are similar.)
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