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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding a remark by Browder after Definition 8.9 ...

The relevant text from Browder reads as follows:

View attachment 9402

At the end of the above text from Browder, we read the following:"... ... Thus Definition 8.9 says roughly that a function is differentiable at \(\displaystyle p\) if it can be approximated near \(\displaystyle p\) by an affine function ... "

My question is as follows: Can someone please demonstrate formally and rigorously that Definition 8.9 implies that a function is differentiable at \(\displaystyle p\) if it can be approximated near \(\displaystyle p\) by an affine function ... ...

Help will be much appreciated ... ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding a remark by Browder after Definition 8.9 ...

The relevant text from Browder reads as follows:

View attachment 9402

At the end of the above text from Browder, we read the following:"... ... Thus Definition 8.9 says roughly that a function is differentiable at \(\displaystyle p\) if it can be approximated near \(\displaystyle p\) by an affine function ... "

My question is as follows: Can someone please demonstrate formally and rigorously that Definition 8.9 implies that a function is differentiable at \(\displaystyle p\) if it can be approximated near \(\displaystyle p\) by an affine function ... ...

Help will be much appreciated ... ...

Peter