Consider $\Omega = (0,2) \subseteq \mathbb{R} $ to be a support of the function $f(x)$ defined as
$$ f(x) = \begin{cases} x, &0 < x \leq 1\\ 2, &1< x \leq 2 \end{cases}$$
I need to compute the first distributional derivative of $f$.
So, I compute the following expression, $$- \int_{0}^{2} f(x) \phi ' (x) \,dx ,$$ for some test function $\phi \in C_{c}^{\infty}(\mathbb{R})$.
The above integral simplifies to
$$-\left( \int_{0}^{1} x\phi ' (x) \,dx + \int_{1}^{2} 2 \phi ' (x)\, dx \right) = \int_{0}^{1} \phi (x) \,dx + \phi(1).$$ using integration by parts.
Here this additional term $\phi(1)$ is bothering me. Can I simply ignore it considering it a constant and write the first distributional derivative $g$ as
$$ g(x) = \begin{cases}
1, &0< x\leq 1\\
0, &\text{otherwise}.
\end{cases}$$
And if this is wrong then what is the correct first distributional derivative?