0
votes
1answer
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 ...
4
votes
3answers
39 views

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 ...
1
vote
1answer
51 views

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 = ...
2
votes
2answers
56 views

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 ...
1
vote
1answer
94 views

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 ...
2
votes
1answer
139 views

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 ...
4
votes
2answers
125 views

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 ...
0
votes
1answer
109 views

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 ...
6
votes
4answers
321 views

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 ...
2
votes
4answers
108 views

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 ...
4
votes
1answer
171 views

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 $\epsilon$-$\delta$ ...
2
votes
1answer
301 views

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 ...
1
vote
2answers
561 views

Evaluation of Derivative Using $\epsilon−\delta$ 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 $\varepsilon$-$\delta$ ...
6
votes
1answer
1k views

Lipschitz Continuous $\Rightarrow$ Uniformly Continuous

The Question: Prove that if a function $f$ defined on $S \subseteq \mathbb R$ is Lipschitz continuous then $f$ is uniformly continuous on $S$. Definition. A function $f$ defined on a set $S ...
3
votes
2answers
410 views

Epsilon-delta proof of continuity

Prove that $f(x,y,z)=x^4+y^4+z^4$ is continuous on point $(x,y,z)=(0,0,0)$ with epsilon-delta I prove this so: if $$\lim_{x,y,z \to 0,0,0} f(x,y,z) = f(0,0,0)$$ then that function is continuous ...
2
votes
2answers
653 views

Proof of Bolzano's Theorem

I know one proof of Bolzano's Theorem, which can be sketched as follows: Set $f$ a continuous function in $[a,b]$ such that ${f(a)<0<f(b)}$. ${A=\{x:a<x<b \text{ and } f <0\in[a,x] ...