What is exactly an element a[i] in mathematica?

Question

I'm confused about the logic behind mathematica programming.

We can have function that will be called doing $f[x]$, thus the element $f[x]$ can be seen as the return value of a function taking an argument $x$ as input.

However, we can also write :

$$ a[1]=2$$

Here, $a[1]$ is not seen as the return value of a function but as a variable named $a[1]$ that has the value $2$.

Now let's take a more tricky example and I do the following :

f[x_] := x^2;

f[1]

1

f[2]

4

f[1] = 100;

f[1]

100

f[2]

4

As we can see, I can replace the value $f[1]$ by $100$, but the function still exists in the end.

My questions :

• What is exactly the quantity $a$ in my example. In another programming language we would call it an array but here it is different.
• What happens in mathematica when I did the replacement $f[1]=100;$ Because it didn't destroyed the function ($f[2]$ still had a value), but in the same time the function doesn't exist for the value $1$. I don't understand what mathematica does exactly. In a lot of other programming languages, this wouldn't be a valid operation this $f[1]=100;$.

Actually my questions are also to understand the philosophy behind the language.