A cubic spline interpolation of a set of data exhibits continuous first and second derivatives throughout the domain of interpolation.

Sure, I can explain this statement step by step.

1. "A cubic spline interpolation of a set of data": This refers to a mathematical method used to approximate data with a smooth curve. The curve is made up of a series of cubic (third degree) polynomials, hence the term "cubic spline".

2. "exhibits continuous first and second derivatives throughout the domain of interpolation": This means that the curve created by the cubic spline interpolation is smooth and has no sharp corners or discontinuities. The first derivative of a function measures its rate of change, or slope. For the curve to be smooth, the slope must change continuously, without any abrupt changes. The second derivative measures the rate of change of the first derivative, or how the slope is changing. For the curve to be smooth, this also needs to change continuously.

3. "throughout the domain of interpolation": This simply means over the entire range of data that is being interpolated. The "domain" of a function is the set of all possible input values (in this case, the data points being interpolated).

So, in simpler terms, this statement is saying that a cubic spline interpolation creates a smooth curve that fits the given data points, with no sharp corners or jumps.