Periodic functions $f(t)$ (with a time period $T$)
can be approximated by a [Fourier *series*][1],
i.e. by summing harmonic oscillations with the *discrete* frequencies
$0, \frac{2\pi}{T}, 2\frac{2\pi}{T}, 3\frac{2\pi}{T}, \ldots$ .
$$f(t)=\sum_{n=-\infty}^{+\infty} F_n e^{in\frac{2\pi}{T}t}$$

But, as you already noticed, with a Fourier series you
cannot build *aperiodic* functions
(e.g. functions limited to a finite range).
For approximating such functions you need a [Fourier *integral*][2],
i.e. summing harmonic oscillations with *all*
frequencies from $0$ to $\infty$.
$$f(t)=\int_{-\infty}^{+\infty}F(\omega)e^{i\omega t}\ d\omega$$

---

### Example 1: Periodic function $\to$ discrete spectrum

As an example for $f(t)$ let's choose the rectangular
wave (with period $T$) extending from $t=-\infty$ to $t=+\infty$.  
[![enter image description here][3]][3]  
$f(t)$ is a perfectly periodic function, and therefore
can be decomposed into a Fourier series.
Its spectrum (the Fourier coefficients $F_n$) look like this:  
[![enter image description here][4]][4]  

### Eaxample 2: Aperiodic function $\to$ continuous spectrum

Now as a second example for $f(t)$ let's choose the rectangular
wave (again with period $T$) restricted to a certain range.
Outside this range we define $f(t)=0$.  
[![enter image description here][5]][5]  
Obviously this function is quite similar to the first example.
Therefore we expect its spectrum to be somehow similar to the
spectrum of the first example.

Because $f(t)$ is aperiodic, it cannot be decomposed into a
Fourier series. But it can be decomposed into a Fourier integral.
Its Fourier spectrum (the function $F(\omega)$) looks like this:  
[![enter image description here][6]][6]  
<sub>(Function plots created with [FooPlot][7])</sub>

This spectrum is continuous, but still quite similar to the
discrete spectrum of the first eample. It has some pronounced
peaks (at the same frequencies as in the first example).
But there is also some spectral intensity outside of these peaks.


  [1]: https://en.wikipedia.org/wiki/Fourier_series
  [2]: http://www.math.toronto.edu/courses/apm346h1/20151/L15.html
  [3]: https://i.sstatic.net/CkESE.png
  [4]: https://i.sstatic.net/YhUhh.png
  [5]: https://i.sstatic.net/UZs1u.png
  [6]: https://i.sstatic.net/z7U4x.png
  [7]: http://fooplot.com