  We prove a finite smooth version of the entropic continuity of Lyapunov
exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9].
As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism
$f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$
admits a measure of maximal entropy. We also prove the $C^r$ continuity of the
topological entropy at $f$.
