Composites of continuous functions Figure A.30 gives the diagram for a proof that the co...

Composites of continuous functions Figure A.30 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 f is continuous at x = c and g is continuous at f(c), then g o f is continuous at c.

Assume that c is an interior point of the domain of f and that f(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.)