We have achieved the simultaneous generation of a 2.6-octave-wide supercontinuum (SC) spectrum over 400-2500 nm and third-harmonic light solely by a dispersion-controlled silicon-nitride waveguide (SiNW). To increase the visible intensity of the SC light component, we fabricated low-loss 5-mm-long deuterated SiNWs with spot-size converters by low-temperature deposition. We succeeded in measuring the carrier-envelope-offset (CEO) signal with a 34-dB signal-to-noise ratio because this short deuterated SiNW provides a large temporal overlap between the f and 3f components. In addition, we have demonstrated this method of CEO locking at telecommunications wavelengths with f-3f self-referencing generated solely by the SiNW without the use of highly nonlinear fiber and an additional nonlinear crystal. Compared with the method of CEO locking with a highly nonlinear fiber and a standard f-2f self-referencing interferometer, this method is not only simple and compact but also stable.