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### When is uniform continuity and continuity same ? [duplicate]

Possible Duplicate: Continuous function on a compact metric space is uniformly continuous How does uniform continuity and continuity coincide in a Compact set ?
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### Why is a continuous mapping from a compact metric space to another metric space is uniformly continuous? [duplicate]

Why is a continuous mapping from a compact metric space to another metric space is uniformly continuous? This theorem is from Rudin Real Analysis Page 202.
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### Does continuity always imply integrability?

Please correct me if I'm wrong. In terms of Riemann integrability: If we are taking into consideration Riemann integrals on a closed interval, then any continuous function is integrable. In terms of ...
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### A continuous function on $S^1$- unit circle .

$$S^1=\{z\in \mathbb C : |z|=1\}$$ be the unit circle. Then which of the following is false $?$ Any continuous function from $S^1$ to $\mathbb R$ is A. bounded B. uniformly continuous. C. has ...
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### Give an example of a function that is bounded and continuous on the interval [0, 1) but not uniformly continuous on this interval.

My thoughts was to take $f(x) =\cos(\frac 1x)$ for all $x \in [0,1)$ as I know this function is continous from $[0,1)$ and is definitely not uniformly continuous as it oscilates non-uniformly. My ...
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### Proof of uniform continuity of a function

Show that the function $f(x) = \cfrac{x^2 + 5x - 7}{(x^2 - 9x + 8)(x-2)}$ is uniformly continuous on the interval $(3,5)$ (not with epsilon and delta) How do I do this question? I am sitting an exam ...
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### Continuous with compact support implies uniform continuity

This might be a duplicate but I tried googling the MSE site and could not find a satisfactory answer. Let $(X, d)$ be a metric space and $f$ be a real valued continuous function on $X$. Suppose $f$ ...
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### Using dominated convergence to prove partial derivative and integral can be interchanged

Hi guys doing a self study here, and came across this problem. I know the same question with slightly different hypotheses has been asked before but I was a bit confused on the answers given and not ...
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### If $f$ is continuous on $[a,b]$ then $f$ is uniformly continuous on $[a,b]$.

So I want to prove that continuity on $[a,b]$ implies uniform continuity with only using the least upper bound property of the reals. I know the basic idea of this, but am getting confused with ...
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### If $f$ is continuous on $\left[ a,b\right]$ then $f$ is uniformly continuous on $\left[ a,b\right]$. [duplicate]

Let $\left[ a,b\right]\rightarrow \mathbb{R}$. If $f$ is continuous on $\left[ a,b\right]$ then $f$ is uniformly continuous on $\left[ a,b\right]$. Proof-trying. Assume $f$ on $\left[ a,b\right]$. ...
### Is $f(x,y)=\frac{1}{x^2+y^2+1}$ uniformly continuous?
Is \begin{align*} f(x,y)=\frac{1}{x^2+y^2+1} \end{align*} uniformly continuous? I was able to show that $f$ has a global maximum at $f(0,0)=1$, but I can't seem to work out a proper estimate for ...