A novel continuous-aperture-array (CAPA)-aided integrated sensing and communication (ISAC) framework is proposed. Specifically, an optimal continuous ISAC waveform is designed to form a directive beampattern for multi-target sensing while suppressing the multi-user interference (MUI). To achieve the goal of optimal waveform design, the directional beampattern of CAPA is first derived based on Green's function, whereafter a reference sensing waveform is obtained through wavenumber-domain optimization. Based on the reference sensing waveform, a weighted functional programming on the tradeoff between sensing beampattern mismatch and MUI is formulated. To solve the resulting problem, an optimal CAPA-ISAC waveform structure is analytically derived using a Lagrangian-transformation and calculus-of-variations method, where the Lagrangian multiplier associated with the optimal waveform structure is determined via Bisection search. The obtained optimal waveform reveals that it is concurrently affected by the reference sensing waveform, the channel correlations and the channel-symbol correlations. Finally, numerical results validate the effectiveness of the proposed system and waveform design, demonstrating that CAPA can achieve significant performance gains against the ISAC designs based on conventional spatially discrete array in both sensing accuracy and communication reliability.