  We study fractions associated to Ford circles which are extracted by means
continuous curves. We show that the extracted fractions have similar properties
to Farey sequences, like the Farey sum, and we prove that every ordered
sequence that satisfies the Farey sum and has two adjacent fractions, can be
extracted from Ford circles through continuous curves. This allows us to relate
sequences of fractions that satisfy the Farey sum and continuous curves. We
focus on the fractions $F_{1/m}$ extracted from inclined lines with positive
slopes of the form $1/m$ and define jumps as the cardinality increments of
these fractions with respect to $m$. We relate the expression for every jump to
the prime omega function in terms of $m$ and find a cardinality formula related
to the M\"obius function, which we approximate with three tractable expressions
that grow in a log-linear way considering estimations of sums related to
Euler's totient function or the graph of the lattice points corresponding to
the fractions $p/q$.
