In general, InterpolatingFunction objects are continuous but not smooth, as in the following example.

It can be clearly seen that these InterpolatingFunction objects are piecewise polynomial functions.

If the derivatives of the function are also available, one can use FunctionInterpolation as follows.

The resulting InterpolatingFunction object is continuous and smooth.

In general, the derivatives of an InterpolatingFunction object are continuous up to the minimum order which is contained in its data.