  The invariant differential cross section for inclusive electron production in
$p + p$ collisions at $\sqrt{s} = 200$~GeV has been measured by the PHENIX
experiment at the Relativistic Heavy Ion Collider over the transverse momentum
range $0.4 \le p_T \le 5.0$~GeV/$c$ in the central rapidity region ($|\eta| \le
0.35$). The contribution to the inclusive electron spectrum from semileptonic
decays of hadrons carrying heavy flavor, {\it i.e.} charm quarks or, at high
$p_T$, bottom quarks, is determined via three independent methods. The
resulting electron spectrum from heavy flavor decays is compared to recent
leading and next-to-leading order perturbative QCD calculations. The total
cross section of charm quark-antiquark pair production is determined to be
$\sigma_{c\bar{c}} = 0.92 \pm 0.15 {\rm (stat.)} \pm 0.54 {\rm (sys.)}$~mb.
