420 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 ?
236 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.
4k 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 ...
2k 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 ...
2k 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 ...
423 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 ...
2k 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 $f$ ...
897 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 ...
783 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 ...
358 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 $\left[ a,b\right]$. ...
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 ...