This paper presents an adaptive neural control to solve the tracking problem of a class of pure-feedback systems with non-differentiable non-affine functions in the presence of unknown periodically time-varying disturbances. To handle with the design difficulty from non-affine structure of pure-feedback system, a continuous and positive control gain function is constructed to model the periodically disturbed non-affine function as a form that facilitates the control design. As a result, the non-affine function is not necessary to be differentiable with respect to control variables or input. In addition, the bounds of non-affine function are unknown functions, and some appropriate compact sets are introduced to investigate the bounds of non-affine function so as to cope with the difficulty from these unknown bounds. It is proven that the closed-loop control system is semi-globally uniformly ultimately bounded by choosing the appropriate design parameters. Finally, comparative simulations are provided to illustrate the effectiveness of the proposed control scheme.