Final answer:
To find values of x at which a function is differentiable, we must ensure that the function is continuous and its first derivative is continuous within the interval in question. Exceptions occur at points where the potential is infinite.
Step-by-step explanation:
To find all values of x for which a function is differentiable, we must ensure that the function meets certain criteria. The function must be continuous, and its first derivative concerning x, dy(x)/dx, must also be continuous. The only exceptions to this are points where the potential function V(x) is infinite, which are generally not considered for differentiability.
In the context provided, if a function is represented graphically by a curve, as in EXAMPLE B11, we need to look at the shape of the curve to determine its differentiability. For instance, a horizontal line segment within an interval, say from 0 to 20, indicates the function is constant and therefore differentiable throughout that interval, as it is smooth and has no breaks or corners.