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I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the whats and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however.

Number 1.

"Let f be a continuous function on the real numbers. Then the set {x in R : f(x)>c} is open for any c in R.

Proof:

Let:

[tex]E_c = \{x \in \mathbf{R}:f(x)>c \}.[/tex]

Assume that x is in E sub c, we have f(x)>c since f is continuous at x there exists delta such that:

[tex] |f(x) - f(y)|<f(x)-c[/tex]

whenever

[tex] |x-y|< \delta.[/tex]

For such y we have:

[tex]f(y) \ge f(x) - |f(x) - f(y)| > f(x) - (f(x) - c)=c[/tex]

and y is in E sub c. We have shown that

[tex](x- \delta, x + \delta) \subset E_c[/tex] and there the set is open."

I get why the set is open and the algebra in the previous step. However I am wondering: Are we kind of setting f(x)-c as our epsilon? And to get to the second last line, are we using some kind of triangle inequality?

2.

"Let f be a function on X, for each positive delta we define:

[tex]\omega_{f,X}(\delta)= sup \{|f(x) - f(y)|: x,y \in X, |x-y|<\delta \}.[/tex]

f is uniformly continuous on X (1) if and only if [tex]lim_{\delta \to 0} \omega_{f,X}(\delta)=0. (2)[/tex]

Proof:

By definition, f is uniformly continuous on X if for any epsilon greater than zero there exists [tex]\delta(\epsilon)>0[/tex] such that [tex]\omega_{f,X}(\delta(\epsilon))<\epsilon.[/tex]

Now, assume (2). Then for any epsilon greater than zero there exists [tex]\delta(\epsilon)>0[/tex] such that [tex]\omega_{f,X}(\delta(\epsilon))<\epsilon[/tex] and f is uniformly continuous.

Assume (1). Then for any epsilon greater than zero there exists [tex]\delta(\epsilon)>0[/tex] such that [tex]\omega_{f,X}(\delta(\epsilon))<\epsilon.[/tex]

Clearly, omega is greater than or equal to zero and non-increasing. We have:

[tex]\omega_{f,X}(\delta)<\epsilon[/tex]

for any [tex]\delta \le \delta(\epsilon)[/tex] and [tex]lim_{\delta \to 0} \omega_{f,X}(\delta)=0[/tex] which we were supposed to prove."

I get a bit confused about the delta functions. What does those mean? In proving (2)=>(1) how can we "guarantee" that there exists such deltas?

In the last step of (1)=>(2), since it is non-increasing, should there not be [tex]\omega_{f,X}(\delta) \le \epsilon[/tex], since [tex]\delta \le \delta(\epsilon).[/tex]

Or am I missing a point along the way?

Thanks.

Edit:

Why can't I have LaTeX coding and regular text in the same paragraph? It looks horribly messy.

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# Proofs with epsilon delta (real analysis)

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