# Increasing Function and Discontinuities ... Browder, Proposition 3.14 ... ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.14 ...

Proposition 3.14 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... For any $$\displaystyle d \in I, d$$ not an endpoint of $$\displaystyle I$$ we know (Proposition 3.7) that $$\displaystyle f(d-)$$ and $$\displaystyle f(d+)$$ exist with $$\displaystyle f(d-) \leq f(d) \leq f(d+)$$, so $$\displaystyle d \in D$$ if and only if $$\displaystyle f(d-) \lt f(d+)$$. ... ... "

My question is as follows:

Can someone please demonstrate (rigorously) exactly why/how it follows that $$\displaystyle d \in D$$ if and only if $$\displaystyle f(d-) \lt f(d+)$$. ... ...

Help will be much appreciated ...

Peter

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The above post mentions Browder Proposition 3.7 ... so I am providing the text of that proposition ... as follows:

Hope that helps ...

Peter

Last edited:

#### Peter

##### Well-known member
MHB Site Helper
Re: Increasing Function and Discontinuitiesl ... Browder, Proposition 3.14 ... ...

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

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.14 ...

Proposition 3.14 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... For any $$\displaystyle d \in I, d$$ not an endpoint of $$\displaystyle I$$ we know (Proposition 3.7) that $$\displaystyle f(d-)$$ and $$\displaystyle f(d+)$$ exist with $$\displaystyle f(d-) \leq f(d) \leq f(d+)$$, so $$\displaystyle d \in D$$ if and only if $$\displaystyle f(d-) \lt f(d+)$$. ... ... "

My question is as follows:

Can someone please demonstrate (rigorously) exactly why/how it follows that $$\displaystyle d \in D$$ if and only if $$\displaystyle f(d-) \lt f(d+)$$. ... ...

Help will be much appreciated ...

Peter

=======================================================================================

The above post mentions Browder Proposition 3.7 ... so I am providing the text of that proposition ... as follows:

Hope that helps ...

Peter

After a little reflection I have realized that $$\displaystyle f$$ is continuous at any point $$\displaystyle x$$ if and only if $$\displaystyle f(x-) = f(x+)$$ ... so if f is discontinuous at $$\displaystyle d$$ then $$\displaystyle f(d-) \neq f(d+)$$ ... but ... we have that $$\displaystyle f(d-) \leq f(d+)$$ ... so therefore $$\displaystyle f(d-) \lt f(d+)$$ ...

Is that correct?

Peter