Confusion about continuity in metric spaces

[gottfried]

I'm confused about the the definition of a function not being continuous.

Is it correct to say f(x) is not continuous at x in the metric space (X,d) if
$\exists$ε>0 such that $\forall$$\delta$ there exists a y in X such that d(x,y)<$\delta$ implies d(f(x),f(y))>ε

Is y dependent on $\delta$? It seems to me as though it should be.

An example let f:(ℝ,de)→(ℝ,de) be defined as f(x)=x if x≤0 and f(x)=1+x if x>0 and prove that f(x) is not continuous at x=0.

So $\exists$ε=0.5>0 such that $\forall$$\delta$ there exists a y=$\delta$ in ℝ such that d(x,y)<$\delta$ implies d(f(x),f(y_δ))>0.5

Is my understanding and implementation of the definition correct? To prove the lack of continuity do I have to find a specific y value which is dependent on delta.

 

[Dick]

I'm confused about the the definition of a function not being continuous.

Is it correct to say f(x) is not continuous at x in the metric space (X,d) if
$\exists$ε>0 such that $\forall$$\delta$ there exists a y in X such that d(x,y)<$\delta$ implies d(f(x),f(y))>ε

Is y dependent on $\delta$? It seems to me as though it should be.

An example let f:(ℝ,de)→(ℝ,de) be defined as f(x)=x if x≤0 and f(x)=1+x if x>0 and prove that f(x) is not continuous at x=0.

So $\exists$ε=0.5>0 such that $\forall$$\delta$ there exists a y=$\delta$ in ℝ such that d(x,y)<$\delta$ implies d(f(x),f(y_δ))>0.5

Is my understanding and implementation of the definition correct? To prove the lack of continuity do I have to find a specific y value which is dependent on delta.

Yes, you've got it. Find an ε such that for all δ there is a y value that violates the conditions of continuity.