I forgot to note $\left|\ell\cap[a,b]\right|=\mathfrak{c}$. I can't make more edits or I could risk being suspended.

What does "$\ell'\to \ell$" mean? Also, per one of your linked posts, there is no hyper discontinuous function on any interval $[a,b]$, and so this is probably not worth considering when building this metric.

Also, what role do the sets $X$ and $Y$ play in your definition? It seems like $f$ has to be a function from $\mathbb{R}$ to $\mathbb{R}$ in order for $\mathcal{D}$ to be well defined.

$\ell^{\prime}$ is a non-vertical line that interects with $\ell$, and rotates around the interesection with $\ell$ to approach the line $\ell$.

Hyper-discontinuous functions cannot be defined on the interval $[a,b]$ but we can take an "almost vertical" $\ell$ in $[a,b]$ to interesect with the hyperdiscontinuous function. Note $[a,b]\cap\mathbb{Q}\cap[0,1]$ does not have to be empty.

I feel like I had to remention $f:X\to Y$ from my post to make the answer clear. It's something I learned from a former professor.

What happens if $X$ doesn't contain any intervals with positive length? Then $f|_{[a,b]}$ isn't well defined for any $a<b$, e.g. if $X=\mathbb{Q}$.

Try and salvage whatever you can. Ignore the rest.

I'm still not sure what the line $\ell'$ is. Is it a straight line or a curve? If it's a curve, how does your choice of particular curve affect the quantity $\mathcal{D}$?

$\ell^{\prime}$ is a straight line

Are you saying it's a straight line very close to being vertical? Say forming a small angle with $\ell$?

Exactly

Then I am not sure your definition is well defined. I don't think in general there is a line $\ell'$ that is not vertical and also intersects all limit points of $\overline{G(a,b)}$, for instance when $f$ is the indicator function for the rationals.

What I meant to say is when the counting measure $|\ell^{\prime}\cap[a,b]|$ is positive infinity, $\ell^{\prime}$ interesects twice with the indicator function for the rationals restricted to [a,b].

If the measure D doesn't make sense, is there any way to change the measure so it gives everything I want in my answer...

I'm not sure of a way to alter what you've done to get everything you want. I think counting the number of times vertical or nearly vertical lines intersects the closure of the graph of the function is maybe not the best idea. For an example of why, it's possible for $\overline{G(a,b)}$ to be the strip $\mathbb{R}\times [-\varepsilon,\varepsilon]$ for any small $\varepsilon>0$, and so the measure of discontinuity will be infinite even though it oscillates less than $1_\mathbb{Q}$ everywhere.

Oh well...thanks for responding.

One way to make this example is to let $A=\{a_i\}_{i=1}^\infty$ be a countable dense set, and let $f(x)=\varepsilon\sum_{i=1}^\infty 2^{-I}\sin(1/(x-a_i))$

I think my answer gets at what you want, except for what the measure of discontinuity of the Dirichlet function is. However, I left a comment there about how my answer could be modified to accommodate this fact. The only issue there is that fixing this requires one to make a choice that may be disadvantageous for other reasons, see my comment there for details.