9. Consider the following outline to a proof of the Chain Rule for real functions.

Let y = f(x), Yo = f(xo), and observe that y ~ Yo as x ~ Xo. Thus lim go f(x) - 9 0 f(xo) = lim g(f(x)) - g(f(xo)) f(x) - f(xo)

X->Xo X - Xo X->Xo f(x) - f(xo) x - Xo

= (lim g(y) - g(yo)) (lim f(x) - f(xo))

Y->Yo y - Yo X->Xo x - Xo

= g'(yo)f'(xo) = g'(f(xo))f'(xo).

(a) Find the flaw in this argument.

(b) Write down a statement that this argument does prove.