Final answer:
The continuity of a function's inverse is contingent on the function being both continuous and bijective (injective and surjective).
Step-by-step explanation:
The continuity of the inverse of a function is dependent on the continuity of the original function, but there are additional requirements that need to be met. For a function to have a continuous inverse, it must be bijective; that is, it should be both injective (each element of the domain maps to a unique element in the range) and surjective (all elements of the range are mapped to by the domain).
In examples given, such as the equation of continuity in fluid dynamics, the principle illustrates that water's velocity increases when passing through a nozzle due to the conservation of mass. For a function like y(x), to have a continuous derivative dy(x)/dx, y(x) must be continuous. However, if the potential V(x) becomes infinite, the derivative may not be continuous. So while the continuity of y(x) is a prerequisite for the continuity of its derivative, the reverse isn't always true; V(x) can disrupt this continuity.
For example, consider the function f(x) = x^2, which is continuous on its domain. The inverse function f^(-1)(x) = sqrt(x) is also continuous on its domain because the original function is continuous.
However, if the original function is not continuous or is not a bijective function, then the inverse function may not be continuous. For instance, the step function f(x) = {0, x < 0; 1, x >= 0} is not continuous, and its inverse function is not continuous either.