  This paper establishes uniform bounds in characteristic $p$ rings which are
either F-finite or essentially of finite type over an excellent local ring.
These uniform bounds are then used to show that the Hilbert-Kunz length
functions and the normalized Frobenius splitting numbers defined on the
Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz
multiplicity function and the F-signature function. From this we establish that
the F-signature function is lower semi-continuous. We also give a new proof of
the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally
proven by Ilya Smirnov.
