  We give a new definition for a Lyapunov exponent (called new Lyapunov
exponent) associated to a continuous map. Our first result states that these
new exponents coincide with the usual Lyapunov exponents if the map is
differentiable. Then, we apply this concept to prove that there exists a
C0-dense subset of the set of the area-preserving homeomorphisms defined in a
compact, connected and boundaryless surface such that any element inside this
residual subset has zero new Lyapunov exponents for Lebesgue almost every
point. Finally, we prove that the function that associates an area-preserving
homeomorphism, equipped with the C0-topology, to the integral (with respect to
area) of its top new Lyapunov exponent over the whole surface cannot be
upper-semicontinuous.
