Let C denote the Banach space of continuous real-valued functions on [0, 1] with the uniform norm. The present article is devoted to the structure of the sets in which the graphs of a residual set of functions in C intersect with different straight lines. It is proved that there exists a residual set A in C such that, for every function/ e A, the top and the bottom (horizontal) levels of / are singletons, in between these two levels there are countably many levels of / that consist of a

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... perfect set together with a single isolated point, and the remaining levels of / are all perfect. Moreover, the levels containing an isolated point correspond to a dense set of heights between the niinimum and the maximum values assumed by the function. As for the levels in different directions, there exists a residual set B in C such that, for every function / e B, the structure of the levels of / is the same as above in all but a countable dense set of directions, and in each of the exceptional nonvertical directions the level structure of /is the same but for the fact that one (and only one) of the levels has two isolated points in place of one. For a general function/ e C a theorem is proved establishing the existence of singleton levels of /, and of the levels of / that contain isolated points.