A Strongly Semismooth Integral Function and Its Application

As shown by an example, the integral function f :ℝn → ℝ, defined by f(x) = ∫ab[B(x, t)]+g(t) dt, may not be a strongly semismooth function, even if g(t) ≡ 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ℝn. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g ≢ 0 in [a, b], and n ≥ 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.