On Rudin's book, there is a comparison between continuity and uniform continuity. It says that if $f$ is continuous on $X$, then it is possible to find, for each $\epsilon>0$ and for each point $p$ of $X$, a number $\delta>0$ having the property specified in DEFINITION of CONTINUITY, and this $\delta$ depends on $\epsilon$ and $p$. For the uniform circumstance you can find a $\delta>0$ which will do for all points of the set. Well I couldn't understand what does $\delta$ depends on $\epsilon$ and $p$ means and why we can find a number $\delta$ for the uniform circumstance. The only difference I could notice is that the uniform continuity is for the whole metric space and the continuity is for a subset, even a point.
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See example of Uniform continuity. – Mauro ALLEGRANZA Feb 28 '23 at 13:27
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2Can you tell me a number larger than $3$? Sure, you could give the answer of $3+1=4$. Can you tell me a number larger than the number I am thinking of right now? Well... if you were to guess a number and say $10$ billion and one, you might be wrong. But... if I were to tell you the number I was thinking of, then you could tell me a number larger than it, for example by adding $1$ to it. That is what we mean by "$\delta$ depends on $\epsilon$." That we can pick a valid number with our desired properties if we were allowed to look at the other number first and use it. – JMoravitz Feb 28 '23 at 13:29
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Good explaination! I suddenly figure it out! – Beginner Mar 01 '23 at 02:29
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As an example the (Real) function $y=\frac 1x$ is continuous in the interval $(0,1]$ but not uniformly continuous. Because a closed interval has a property called compactness, continuous functions on closed intervals are always uniformly continuous.
As with all definitions the main question is "how is this useful?" and you might find it helpful to see how the property is used in order to understand why the definition is useful.
Mark Bennet
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