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

I am focused on Chapter 5: Continuous Functions ...

I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...

Theorem 5.3.2 and its proof ... ... reads as follows:

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

" Since $ \displaystyle I$ is closed and the elements of $ \displaystyle X'$ belong to $ \displaystyle I$, it follows from Theorem 3.2.6 that $ \displaystyle x \in I$. Then $ \displaystyle f$ is continuous at $ \displaystyle x$ ... ... "

Can someone please explain exactly why/how we can conclude that $ \displaystyle f$ is continuous at $ \displaystyle x$ ... ?

Peter