Hi J. Rafiee

thanks for the explanation and following links,

Regarding my question, given the output drawn for wavelet functions, what does it mean to obtain the value of functions at a particular point in wavelet transform?

Can a wavelet and scale function be a sequence that converges to a point or not?

Values of scaling and wavelet function can be computed on the dense subset of dyadic numbers via the cascade algorithm. As that has the nature of an iterated function system, one could also play a chaos game to get a collection of values over random sample points.

I'll attach a file with a longer explanation with some exploratory code.

Note that your interpretation of the coefficients is wrong. The c-sequence that you cite sums up to one, in the multi-scale formula the factor has thus to be 2 instead of the square root.

The mother wavelet and the scaling functions are continuous in time functions. Any continuous in time finite energy signal can be decomposed into a wavelet series, using translations and dilations of the scaling function and of the mother wavelets. The sequences of the corresponding coefficients form the result of the discrete wavelet transform. So, strictly speaking, a wavelet and scale function can not be a sequence that converges to a point.