Question: Recently I came with the problem: define a metric for a space formed by nouns.
Here is my formulation:

Let $W$ be the set of all words and let $S \subset W$ be the set of all nouns. Let $A$ be the set of sequences from $W$ such that, for each element $a \in A$, the range of $a$ contains at least one element from $S$.

Define the function $N: S \times S \rightarrow \mathbb{Z}$ given by $N(x, y)$ = the number of sequences on A that has both $x \in S$ and $y \in S$ in its range, where $\mathbb{Z}$ is the set of all integers. Define the map $dep: S \times S \rightarrow \mathbb{R}\_+$ such that $dep(x,y) = N(x,x) - N(x,y)$, where $\mathbb{R}\_+$ is the set of all non-negative real numbers.

Question: is $dep$ a metric? if not, is it a quasimetric?

Answer: 
Set $x+t=u$, so that $t=a$ means $u=x+a$ and $t=-a$ means $u=x-a$. Then
$$
F(x)=\frac{1}{2a}\int\_{x-a}^{x+a} f(u)\, du.
$$
You can now apply the fundamental theorem of calculus.
You are given that $f(x)$ is continuous, but you are not given that $f\_{x}(x,t)$ is continuous which is a requirement in Leibniz rule. The rule for interchanging differentiation with integration the way you did. So just follow Siminore's answer.