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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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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 $F\colon [a,b]\... 3answers 2k views ### Does continuity always imply integrability? Please correct me if I'm wrong. In terms of Riemann integrability: If we are taking into consideration Riemann integrals on a closed interval, then any continuous function is integrable. In terms of ... 3answers 1k views ### A continuous function on$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 ... 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 ... 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)$and is definitely not uniformly continuous as it oscilates non-uniformly. My ... 2answers 1k views ### Continuous with compact support implies uniform continuity This might be a duplicate but I tried googling the MSE site and could not find a satisfactory answer. Let$(X, d)$be a metric space and$f$be a real valued continuous function on$X$. Suppose$f$... 1answer 678 views ### Using dominated convergence to prove partial derivative and integral can be interchanged Hi guys doing a self study here, and came across this problem. I know the same question with slightly different hypotheses has been asked before but I was a bit confused on the answers given and not ... 2answers 171 views ### If$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]$. ... 3answers 88 views ### Is$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 thatf$has a global maximum at$f(0,0)=1$, but I can't seem to work out a proper estimate for ... 1answer 368 views ### If$f$is continuous on$[a,b]$then$f$is uniformly continuous on$[a,b]$. So I want to prove that continuity on$[a,b]$implies uniform continuity with only using the least upper bound property of the reals. I know the basic idea of this, but am getting confused with ... 0answers 402 views ### Proof of uniform continuity on compact sets Show that a function$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$and let$...
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 ...
### Prove that $x^n e^{-x}$ is uniformly continuous on $[0, \infty)$
Thus far I have shown that $|f(x)-f(y)| = e^{-x} |x^n - y^n e^{x-y}|$