In the QFT book by Itzykson and Zuber, the variation of the action functional $I=\int_{t_1}^{t_2}dtL$ is written as:

$$\delta I=\int_{t_1}^{t_2}dt\frac{\delta I}{\delta q(t)}\delta q(t)$$

How is this justified? In particular, why is there an integration sign?