Discreteness and periodicity in Fourier transform
This is, essentially, what the sampling theorem is about.  Uniform sampling in one domain (e.g. the time domain) causes periodic extension in the reciprocal domain (e.g. frequency domain).
The reason why is that the sampling function is a periodic function which means it can be represented as a Fourier series
$$\begin{align} 
\operatorname{III}_T(t) \ &amp;\triangleq\  \sum_{k=-\infty}^{\infty} \delta(t - k T) \\
 &amp;= \sum_{n=-\infty}^{+\infty} c_n e^{j 2 \pi n \frac{t}{T}}  \\
\end{align}$$
where the Fourier coefficients are
$$ \begin{align}
c_n\, &amp; = \frac{1}{T} \int_{t_0}^{t_0 + T} \operatorname{III}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \quad ( -\infty &lt; t_0 &lt; +\infty ) \\[4pt]
&amp; = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \operatorname{III}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \\[4pt]
&amp; = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \delta(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \\
&amp; = \frac{1}{T} e^{-j 2 \pi n \frac{0}{T}} \\[4pt]
&amp; = \frac{1}{T} \ .
\end{align} $$
Using the definition of Fourier Transform most common with electrical engineering
$$ X(f) \triangleq \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ \mathrm{d}t $$
$$ x(t) = \int\limits_{-\infty}^{+\infty} X(f) e^{+j 2 \pi f t} \ \mathrm{d}f $$
when sampling $x(t)$
$$\begin{align}
x_\text{s}(t) &amp;\triangleq x(t) \cdot T \cdot \operatorname{III}_T(t) \\
 &amp;= x(t) \cdot T \sum_{k=-\infty}^{\infty} \delta(t - k T) \\
 &amp;= T \sum_{k=-\infty}^{\infty} x(t) \ \delta(t - k T) \\
 &amp;= T \sum_{k=-\infty}^{\infty} x(kT) \ \delta(t - k T) \\
\end{align}$$
which shows how $x(t)$ is converted to samples $x(kT)$ and this is also true:
$$\begin{align}
x_\text{s}(t) &amp;\triangleq x(t) \cdot T \cdot \operatorname{III}_T(t) \\
 &amp;= x(t) \cdot T \cdot \sum_{n=-\infty}^{+\infty} \frac1T e^{j 2 \pi n \frac{t}{T}} \\
 &amp;= \sum_{n=-\infty}^{+\infty} x(t) \ e^{j 2 \pi n \frac{t}{T}} \\
\end{align}$$
the resulting Fourier transform is
$$ X_\text{s}(f) = \sum_{n=-\infty}^{+\infty} X\left(f - n\tfrac{1}{T} \right)
 $$
which is periodic in the frequency domain with period $\frac{1}{T}$.
Because of the Duality of the Fourier Transform, if can be also shown that sampling in the frequency domain causes periodicity in the time domain, which is essentially all that Fourier series is about.