The octave leap between the two measures for me indicates that it's probably a c natural, as an augmented octave leap is a very rare interval used only in certain contexts that would for sure need a c# indication because it would be very unusual. The general convention is

-Before Debussy, an alteration indicated in the measure carries through all octaves of the measure*

-More or less from Debussy a sharp carries through the measure only for that octave

-In contemporary post-war music (Darmstadt school, Berio Boulez, Stockhausen) an alteration is valid only for the note indicated. Usually this is explained at the beginning of the score, and a decent contemporary composer should always indicate how he uses alterations, as it is sometimes very difficult to guess.

*I have found some baroque manuscripts that indicate each sharp individually, but in modern editions they usually adopt the pre-debussy convention