syms x_1 x_2 x_int x x_hk sigma
h(x_1, x_2) = sign(x_1-x_2)*sqrt(abs(x_1-x_2))
h(x_int) = subs(h(x_1, x_2), x_1-x_2, x_int)

heat_kernel(x_hk, sigma) = exp(-(x_hk)^2/(2*sigma^2))/(sqrt(2*pi*sigma^2))
assume(sigma, {'real', 'positive'})
g(x, sigma) = int(heat_kernel(x-x_int, sigma)*h(x_int), x_int, -inf, inf)

fplot(h(x_int), [-0.4, 0.4])
hold on
fplot(g(x, 0.05), [-0.4, 0.4])
fplot(g(x, 0.1), [-0.4, 0.4])
fplot(g(x, 0.01), [-0.4, 0.4])
legend(["h(x)", "g(x, 0.05)", "g(x, 0.1)", "g(x, 0.01)"], 'Location', 'NorthEastOutside')