Continuous Function: Showing f is Continuous

[CarmineCortez]

I have an assignment question

" let (X,d) be a metric space. a is element of X. Define a function f maps X -> R by f(x) = d(a,x). show f is continuous."

I'm not sure what this function looks like. Is f(x) = sqrt(a^2+x^2) and if it is I need abs(x-a) < delta?? I'm confused.

 

[quadraphonics]

I'm not sure what this function looks like. Is f(x) = sqrt(a^2+x^2)

That would be one possibility, but the problem, as written, is not specific to a particular metric. You need to show the result for an arbitrary metric. Note that all metrics have the following 4 properties:

1) $d(x,y) \geq 0$
2) $d(x,y) = 0$ if and only if $x = y$
3) $d(x,y) = d(y,x)$
4) $d(x,z) \leq d(x,y) + d(y,z)$

and if it is I need abs(x-a) < delta?? I'm confused.

What you want to show is that if $|f(x) - f(y)| < \epsilon$, then there exists some $\delta$ such that $d(x,y) < \delta$. Given the construction of $f(x)$ used here, I would expect property 4 (aka the Triangle Inequality) to be useful here.

Sorry - LaTeX rendering seems to be broken. Please refer to the underlying text in the meantime (hit quote and you will see it).

 

[CarmineCortez]

I think I have it now, is delta = epsilon? from the triangle inequality? if d(a,x),d(a,x_o) < epsilon/2