I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 6: Differentiation ...

I need help in fully understanding an aspect of the proof of Theorem 6.1.6 ...

Theorem 6.1.6 and its proof ... ... reads as follows:

In the above text from Bartle and Sherbert we read the following:

"... Since the function $ \displaystyle ( \psi \circ f ) \cdot \phi$ is continuous at $ \displaystyle c$, and its value at $ \displaystyle c$ is $ \displaystyle g' (f (c) ) \cdot f'(c)$ , Caratheodory's Theorem gives (11) ...

Could someone please explain exactly how Caratheodory's Theorem gives (11) ...?

Peter

*** NOTE ***

The post above mentions Caratheodory's Theorem ... so I am proving the text of the theorem statement ... as follows: