Wavelet transforms have finite duration basis functions, so they can be integrated from -inf to +inf (which essentially reduces to the finite support of the basis functions).
Fourier transforms have infinite duration (also infinite energy) basis functions, so a continuous fourier transform cannot be computed in general for all signals.
Also, only the basis functions are continuous in CWT; the result is an array of coefficients. The output for a CFT is also continuous and not a discrete array of numbers.
You can definitely compute CFT of signals for a specific frequency point in MATLAB.
