Is this a linear functional on $H^1$?

Question

Let $a$, $b \in \mathbb{R}$, $f \in L^2(a,b)$ and define the map: $\varphi:H^1(a,b) \to \mathbb{R}; v \mapsto \int_a^b f(x) \cdot v(x) ~dx$.
Is $\varphi$ now a functional on $H^1(a,b)$?
Clearly it is linear but I don't see why it should be continous.
I would need to find $L>0$ with $\mid \int_a^b f(x) \cdot v(x) ~dx \mid \leq L \left\lVert v \right\rVert_{H^1} $ but I don't see an obvious candidate.