Is the value of a frequency bin of a DFT-output the average of the 'real' frequency values within that bin's range?
I'll start with a counter-example:

Assume your continuous-time signal is a sinusoid at freqeuncy $f_0$. You then sample that signal, and use a rectangular window to select $N$ samples. Feed those $N$ samples to your DFT, and you will see non-zero values in the whole spectrum (unless in the special cases where $f_0$ is a multiple of $f_s/N$).

So you start with a signal whose energy is concentrated in a single frequency, and you end up with non-zero values in all DFT bins.

To understand what is happening:


Start with a continuous-time signal $x_c(t)$.
(Optionally) filter the signal to make it band-limited to $f_s/2$.
Sample the signal to get $x[n] = x_c(n/f_s)$.
At this point, the spectrum is given by the sampling theorem. You will have spectrum aliasing if the sampled signal was not band-limited in step 2.
Now apply a window of size M. The window type depends on your application (Rectangular (ugly), Hann, Hamming, Blackman, Flat top, etc.).
At this point, your spectrum is that of the sampled signal, but convolved with the window (depending on window size and type, you will get some energy spread to other frequencies). This spectrum is $X_w(e^{j\theta})$.
The DFT of the windowed signal will return samples of its spectrum, at frequencies $Y[k] = X_w( e^{j 2\pi k/N})$.