C

Step 1: Analyze Statement A
The characteristic X-ray spectrum does indeed depend on the nature of the material of the target. Different materials have different atomic structures, which means they will emit X-rays at specific energies unique to those elements when bombarded by electrons. Therefore, Statement A is true.

Step 2: Analyze Statement B
The short wavelength limit of the continuous X-ray spectrum is determined by the maximum energy of the electrons in the X-ray tube, which is proportional to the potential difference (V) applied across the tube. Specifically, the minimum wavelength (\( \lambda_{min} \)) can be related to the applied voltage (\( V \)) by the equation \( \lambda_{min} = \frac{hc}{eV} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( e \) is the charge of an electron. This shows that as the potential difference increases, the minimum wavelength decreases, which implies it varies inversely. Hence, Statement B is also true.

Conclusion: Since both A and B are true, the correct choice is Option C.