Using continuous verses discrete wavelet transform in digital applications
As Mohammad stated already the terms Continuous Wavelet Transforms (CWT) and Discrete Wavelet Transforms (DWT) are a little bit misleading. They relate approximately as (Continuous) Fourier Transform (the math. integral transform) to DFT (Discrete Fourier Transform).

In order to understand the details it is good to see the historical context. The wavelet transform was originally introduced in geophysics by Morlet, and was basically a Gabor transform with a Window that grows and shrinks together with the selected scale/frequency. Later Daubchies (a physicist-ett from Belgium) realized that by choosing special orthogonal wavelet bases the infinitely redundant CWT can be critically sampled on a dyadic grid. From the resulting DWT the corresponding full CWT can be obtained by convolving the DWT with the reproducing kernel of the respective wavelet. The reproducing kernel is the CWT of the wavelet itself.

Daubchies findings gave a big boost to the wavelet theory in the early 80ies. The next big result was that the DWT can be computed very efficiently (this is sometimes called FWT [fast WT] as well) by using techniques from the theory of filterbanks, namely quadrature mirror filters (QMF) together with downsampling filterbanks. By constructing special QMFs the corresponding DWT can be computed via filtering and downsampling, which is the state-of-the-art algorithm to compute DWTs today. You do not need the scaling function to compute the DWT, it is just an implementation detail that FWT process.

Concerning the application side the CWT is the more ideal candidate for signal or time series analysis due to its more fine grained resolution and is usually chosen in most tasks (e.g. singularity detection). The DWT is more of interest in the context of fast non redundant transforms. The DWT has a very good energy compactification and is thus a good candidate for lossy compressions and signal transmissions.



Hope that clarified things.