Scaling Function and Wavelet Function
In my experience, scaling functions are not so central in the CWT. However, there are quite important on the DWT. Scaling functions act like low-pass operators, and wavelets like band-pass ones. Indeed, in the DWT, one seldom uses th wavelet and the scaling function directly, since they are hidden in the construction of the low-pass and high-pass filters that are combined in cascading filterbanks. The actual wavelet (and scaling function) are only obtained as a limiting process involving an infinite cascade of discrete filters.
A complementary discussion: Scaling Function in Wavelet Analysis