# Tagged Questions

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### Proving or disproving uniform continuity of various real functions

Okay before anyone does anything, I don't want any of you guys to just write out proofs for all of these, that's asking a bit much :P Maybe just do one, and make it detailed because I really need ...
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### Question regarding $\epsilon-\delta-$proof

I want to prove the continuity of $f(x) = x^2$. Lets take $\epsilon > 0$ and $|x-x_0| <\delta$. I do: $$|f(x) - f(x_0) |= |x^2 - x_0^2| = |(x-x_0)(x+x_0)| < \delta |x-x_0|$$ Now the ...
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### Uniform continuity of a function and cauchy sequences

So I'm pretty sure this is almost immediate from the definitions, please tell me if I am incorrect.. Consider two cauchy sequences in D, $\{x_n\}$ and $\{y_m\}$. Since $f$ is uniform continuous we ...
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### Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
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### Proof that a complex function is continuous at $z=0$

Given the function $f\colon \mathbb{C}\to\mathbb{C}$ by $f(z)=\begin{cases} \frac{xy(x+iy)}{x^2+y^2} & \text{if } z\neq 0\\ 0 & \text{if } z=0 \end{cases}$ with $z=x+iy$. How do I ...
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### Proving discontinuity

Assume set $A$ is countable and let$$f(x)=\cases{1 \text{ if }x\in A\\0\text{ if }x\notin A }.$$ Prove that $f$ is not continuous at $c\in A$. I've seen such a problem before where ...
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### Continuity of 'vectorial' function $\frac{x^2}{y^2-1}$

given is $f(x,y) = ( \frac{y}{x^2+1}, \frac{x^2}{y^2-1} )$. I have to study the continuity of the function for$(x,y) \to (0,1)$. First function $f_1$ is continuous, since $lim f_1 = 1/1 = 1$ so the ...
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### Proving nondifferentiability at all points of a continuous function

Given: $f_1(x)=x$ if $x\le1/2$ $f_1(x)=1-x$ if $1/2\le x\le1$ $f_1(x+1)=f_1(x)$ $\forall n\ge2,f_n(x)=(1/2)*f_{n-1}(2x)$ Let $S_m(x)=\sum_{n=1}^m f_n(x)$ $S_m$ is a continuous function on ...
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### Example of uniformly continuous function on R that is not differentiable on all of R

Give an example of a uniformly continuous function $g:\mathbb{R} \rightarrow \mathbb{R}$ that is not diff erentiable on all of $\mathbb{R}$. Hmm. I can't think creatively enough for one! Would f(x) = ...
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### Continuity of a function with respect to various topologies (example)

Let $f: \mathbb{R} \to \mathbb{R}$ be given by $$f(x)= \begin{cases} -x-1, &\text{if }xâ‰¥0\\ 1, &\text{if }x<0. \end{cases}$$ Denote by $U$ the usual topology on $\mathbb{R}$. $H$ the ...
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### Function that is onto and continous

Can there be a continuous onto function f : R -> R \ Q? I know if a function existed it would map reals to the irrationals. Any ideas?
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### functional equation, find $h$ continous on $\mathbf{R}$ such that $h(x) + h(2x) + h(4x) = {x^n}$

I am having trouble finding easily $h$ defined and continuous on $\mathbf{R}$ verifying for all $x$ in $\mathbf{R}$, $$h(x) + h(2x) + h(4x) = {x^n}$$ where $n$ is a fixed natural number. I have a ...
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### Give an example of a uniformly continuous function on [0, 1] that is differentiable on (0, 1) but for which f' is not bounded on (0, 1)

Any help would be appreciated! Would $f(x) = \sqrt(x)$ work? Give an example of a uniformly continuous function on [0, 1] that is differentiable on (0, 1) but for which f' is not bounded on (0, 1)
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### Introduction to Analysis - Continuity and sequences

Question: Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function satisfying: $\displaystyle \lim_{x\rightarrow +\infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = -\infty$ part a) Show that ...
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### Real Analysis: Cauchy, continuous sequence examples

a continuous function $f: (0,1) \to \mathbb{R}$ and a Cauchy sequence $(x_n)$ such that $f(x_n)$ is not a Cauchy sequence. a continuous function $f: [0,1] \to\mathbb{R}$ and a Cauchy sequence $(x_n)$ ...
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### Introduction to Analysis: The Riemann Integral

The following is a problem from Arthur Mattuck's book, "Introduction to Analysis." Page 265. Assume $f(x)$ integrable on $I$. Prove $F(x) = \int_a^x f(t)\,dt$ is continuous on $I$ How would I ...
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### Introduction to Analysis: Convexity

A friend and I were trying to figure out this problem from our assignment. Prove that on an open $I$, a geometrically convex function $f(x)$ is continuous. To better assist the audience, it is ...
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### For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$

Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ My ...
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### Pointwise Convergence and Continuity

I am having trouble knowing how to start a homework problem. If anybody could give me the first step, or lead me through a different but similar example, that would be greatly appreciated. The problem ...
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### Why are $g(1) < 0, g(0) > 0$ in proof of Fixed-point property for continuous function on $[0,1]$.

Can someone please help me understand how, in the solution below, we can get to $g(1)<0$ and $g(0)>0$ from the fact that $g(1) \neq 0$ and $g(0) \neq 0$? Thanks. Rudin Chapter 4, question ...
### epsilon-delta proof for continuity if $1/f$
How can I prove that $1/f$ is continuous on $[a,b]$ if $f:[a,b] \rightarrow R$ is continuous on $[a,b]$ and $f(x)$ is never $0$ by an epsilon-delta proof? Thank you.