Herbivores in quest of food have to attend areas where predators also concentrate their attacks. They have to deal with two conflicting demands, the need of feeding and that of avoiding predation. Using a theoretical approach, we study the influence of these antagonistic constraints on the shaping of an herbivore's foraging strategy and their consequences in terms of functional response. In our model, the animal may curtail predation risk by being vigilant while foraging and by limiting its time of presence in the feeding area. Being vigilant means either stopping all feeding activities to scan for predators (exclusive vigilance) or using the time spent handling resources to do so, with reduced detection capacities (routine vigilance). Using an optimization criterium, we thus determine how long the animal will stay in the feeding area and how it will share its time and attention while present there, as a function of resource density and predation risk. All behavioural variables show smooth but sometimes non-monotonic variations in response to changes in these environmental conditions. When these variations are taken into account, the instantaneous intake rate of the herbivore, as well as the total biomass of plant consumed, increase in a decelerating but continuous manner with the patch's richness, in qualitative agreement with a type II functional response. This smooth pattern is explained by the possibility for the herbivore, when food abundance rises, to progressively redirect its attention to routine antipredator vigilance. The relative importance of routine vigilance with respect to exclusive vigilance declines when predation risk increases, as well as the duration of the patch's visit and the total quantity of food eaten. The exact shape of the functional response is thus mediated by subtle adjustments of behavioural and attentional priorities, in response to variations in both food availability and predation risk.