# Tagged Questions

53 views

### Prove that a function is continuous at x =0

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$- proof $$f(x) = \begin{cases} x/(1-x),&x\geq 0 \\ x/(1+x),&x \leq 0 \end{cases}$$ So this is what I have so far: Let ...
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### Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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### Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
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### Proving the continuity of a difference of functions.

Prove that if $f$ and $g$ are continuous at $x=a$, then $(f-g)$ is continuous at $x=a.$ I have $|f(x)-f(a)-g(x)+g(a)| = |(f(x)-g(x))-(f(a)-g(a))|$ so far. I wanted to use the triangle inequality on ...
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### Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ is a continuous function using epsilon-delta.

THE QUESTION: Use the metric $(x,y)$ = $\rho(x,y)=|x-y|$ for the reals and use the metric $\rho((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ for the plane. Define $f:R\times R \to R$ as ...
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### Proving that a holder continuous function always has a smaller exponent.

According to wikipedia if we have $f:X \rightarrow Y$ which is $\alpha$-Holderian then for all $\beta < \alpha$ the function is also $\beta$-Holderian. How do we prove this starting from the fact ...
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### If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. Question on proof.

Let $X,Y$ be metric spaces and $f:X\rightarrow Y$. If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. The text I am using proves this proposition like so: Suppose ...
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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 ...
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### How to show that a limit cannot be another number?

Let: $$G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right.$$ I can understand that the function is continuous at $x=0$ because: For ...
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### Can the definition of continuity be said both of these ways? [duplicate]

So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because ...
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### prove $x \mapsto x^2$ is continuous

I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument. The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$. Given the $\epsilon$-$\delta$ ...
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### prove with $\epsilon$-$\delta$-argument: $x\rightarrow |-2x+3|$ is continuous

i am asked to prove with $\epsilon$-$\delta$-argument that $x\rightarrow |-2x+3|$ is continuous my steps: Definition of $\epsilon-\delta$-argument: $\forall \epsilon >0 \exists \delta>0$ with ...
### Evaluation of Derivative Using $\epsilonâˆ’\delta$ Definition
Consider the function $f \colon\mathbb R \to\mathbb R$ defined by $f(x)= \begin{cases} x^2\sin(1/x); & \text{if }x\ne 0, \\ 0 & \text{if }x=0. \end{cases}$ Use $\varepsilon$-$\delta$ ...