First you need to understand what a periodic function is in mathematics. For example, if you are standing at any point P on a circle, you start walking on the circumference and count the degrees you completed. Once you complete 360 degrees, and you will find yourself on exactly the same point P. Sines, cosines and many other functions are periodic. This term existed before the chemists started using the term periodic law in the early 1800s or perhaps well before that. If $f(x)$ is a function with period $T$, then $f(x+T)= f(x)$ for a periodic function in a defined range of $[a,b]$. The period $T$ may change to $T'$, when x is in a different range say, $[c,d]$. This is an example of a discontinuous periodic function. As you will see, periodic properties of the elements form a discontinuous periodic pattern in a loose sense.
Now come to the periodic law: "The law that the chemical elements, when listed in order of their atomic numbers (originally, atomic weights), fall into recurring groups, so that elements with similar properties occur at regular intervals." [unabridged OED].
Instead of $x$ for a mathematical function, the independent variable is the atomic numbers Z (Zahl for a number in German) for the chemists. So this periodic law has a very loose connection with mathematical periodic function, because none of the values are actually repeated. Each element is unique, but in a loose way, their properties or even a better word is trends "repeat" themselves after a certain number of elements. See this chart for example. Atomic radii vs. Z, do you see a loose periodic pattern?
Once you have got the idea of periodic function in a loose way for the chemical elements, get ready to the see the physicists style of periodic table. It explains why some properties loosely repeat themselves.
Each element in the column, corresponds to filling of a certain orbital, for example look at the column $s^1$. All elements falling in that column correspond to the filling of $s^1$ electrons. Then move to the next column $s^2$, all elements listed there corresponding to filling of $s^2$. Can you see a pattern? In this sense, one calls it periodic function in a modern way, that certain elements repeatedly fall in $s^1$ column, some elements repeatedly fall in $s^2$ column.