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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:View attachment 7277In 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
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:View attachment 7277In 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