# Tagged Questions

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### Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
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### representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0$ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
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### epsilon-delta proof for continuity if $1/f$ [closed]

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.
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### Real Analysis: Continuity of a Composition Function

Suppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$. My idea: Can I go straight from definition ...
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### Introduction to Analysis: Continuity and Limits

My coworker and I were looking at a problem for our Real Analysis class. It reads: Call a function "multiplicatively periodic" if there is a positive number c $\neq$ 1 such that $f(cx) = f(x)$ for ...
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### Proof of continuity [closed]

Let $$f:\mathbb{R}\mapsto \mathbb{R}.$$ Prove that if f is differentiable at a real number c, then f is continuous at c.
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### Prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2.

I am looking to prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2. So we nee that given any $\epsilon>0,\ \exists\delta>0$ so that $|f(x)-f(2)|<\epsilon\\$ whenever ...
How do I prove that $f(x)=e^x$ is a continuous function at the point $x=0$? I understand that anything raised to the $0$ power equals $1$, therefore it is continuous. But I don't know how to write a ...