What's the meaning of the continuity in spectrum analysis? We all know that for any, suitable, kind of signal $f(t)$,  there corresponds a Fourier transform function $F(j\omega)$ such that
$$
f(t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} F(j\omega) e^{i\omega t} d\omega
$$

So, $F(j\omega)$ can be recognized as the component of $f(t)$ along $e^{i \omega t}$. This is similar to orthogonal analysis. Like a cartesian vector $\vec{a} = (1,2,3)$ can be represented as 
$$ 
\vec{a}=  1\hat{x}+2\hat{y}+3\hat{z}
$$
where  $\hat{x}$, $\hat{y}$, and $\hat{z}$ are the orthonormal basis vectors (unit vectors along x-y-z directions). 

However, these two things are different in one perspective.Among $x$,$y$,$z$, there is no concept as 'distance'. For example, in Fourier transform. we can say that the distance between $\omega_1 = 1$ and $\omega_2=2$ is smaller than the distance between $\omega_2$ and $\omega_3 =4$. 

This algebra structure in Fourier transform definitely has some extra meaning, but what is it? Does it mean that the Fourier transform makes "more sense" compared with other orthonormal basis analyses where there is no easily-definable distance between basis functions?
Let $v(x, k)$ be an orthonormal set of basis functions such that
$$
\int_{-\infty}^{+\infty} v(x,a) v^*(x,b) dx = \delta_{a,b}
$$
where $\delta_{a,b}$ s the Kronecker delta.

I believe your question boils down to: for the Fourier Transform, we have
$$
v(x,a) = e^{jax}
$$
and we interpret $a$ as the frequency and $x$ as time; is there anything special about this orthonormal basis because we can interpret $a$ as a "distance" between these functions?

An example of another set of functions that are orthonormal (but over a finite interval) are the Legendre Polynomials:
$$
v(x,a) = P_a(x), \mbox{where} -1\le x \le +1
$$
where now $a$ is the order (degree) of the polynomial.

For these (and other polynomial bases), the "distance" is now the difference in polynomial degree.

Whether this means such functions are "less" special than the Fourier bases is a matter of opinion.