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.
The matlab CWT is discrete wavelet transform and it has orthogonal wavelet functions. But the nonorthogonal is not discrete CWT. So you cannot carry out nonorthogonal CWT (i.e continuous CWT) using inbuilt matlab cwt, but you can carry out orthogonal CWT(discrete CWT) using using matlab inbuilt cwt.