Linked Questions

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1answer
378 views

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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1answer
154 views

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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0answers
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The uniform continuity of functions with Banach space values. [duplicate]

Let $f\colon [a,b]\to\mathbb{R}$ be a continuous function. Since $[a,b]$ is compact, then by continuity of $f$ we also have that $f$ is uniformly continuous on $[a,b]$. Suppose now that $F\colon [a,b]\...
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3answers
2k views

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 ...
5
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3answers
1k views

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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2answers
406 views

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 ...
4
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2answers
783 views

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 ...
5
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2answers
1k views

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 $f$ ...
2
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1answer
678 views

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 ...
3
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2answers
171 views

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 $\left[ a,b\right]$. ...
2
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3answers
88 views

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 ...
4
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1answer
368 views

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 ...
3
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0answers
402 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let $...
3
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2answers
87 views

Does this strengthening of continuity have a characterization in terms of familiar concepts?

Definition 0. Whenever $X$ is a metric space, $A \subseteq X$ is a subset, and $r \in \mathbb{R}_{>0}$ is a positive real number, define that $$A \oplus r = \bigcup_{a \in A} B_r(a).$$ Definition ...
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3answers
228 views

Prove that $x^n e^{-x} $ is uniformly continuous on $[0, \infty)$

Thus far I have shown that $|f(x)-f(y)| = e^{-x} |x^n - y^n e^{x-y}|$

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