Gamma-ray burst (GRB) 120729A was detected by Swift/BAT and Fermi/GBM, and then rapidly observed by Swift/XRT, Swift/UVOT, and ground-based telescopes. It had a single long and smooth gamma-ray emission pulse, which extends continuously to the X-rays. We report Lick/KAIT observations of the source, and make temporal and spectral joint fits of the multiwavelength light curves of GRB 120729A. It exhibits achromatic light-curve behavior, consistent with the predictions of the external shock model. The light curves are decomposed into four typical phases: onset bump (Phase I), normal decay (Phase II), shallow decay (Phase III), and post-jet break (Phase IV). The spectral energy distribution (SED) evolves from prompt gamma-ray emission to the afterglow with photon index from $Gamma_{rm gamma}=1.36$ to $Gamma approx 1.75$. There is no obvious evolution of the SED during the afterglow. The multiwavelength light curves from gamma-ray to optical can be well modeled with an external shock by considering energy injection, and a time-dependent microphysics model with $epsilon_Bpropto t^{alpha_B}$ for the emission at early times, $T < T_{rm 0} + 157$~s. Therefore, we conclude that both the prompt gamma-ray emission and afterglow of GRB 120729A have the same external shock physical origin. Our model indicates that the $epsilon_B$ evolution can be described as a broken power-law function with $alpha_{rm B,1} = 0.18 pm 0.04$ and $alpha_{rm B,2} = 0.84 pm 0.04$. We also systematically investigate single-pulse GRBs in the Swift era, finding that only a small fraction of GRBs (GRBs 120729A, 051111, and 070318) are likely to originate from an external shock for both the prompt gamma-ray emission and afterglow.
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