I may not know how to translate Japanese to English but I can try my hand on this challenge:
Let u be a fixed but arbitrary real number.
Define U to be the subset of the real plane that lies below the line that intersects the axis of abscissas at u and the axis of ordinates at u as well.
Define Ψ to be the function from the real plane to the real plane that:
maps each point at the right side of the axis of ordinates to a point that lies in the same vertical line but drops below its original position a number equal to its distance to the axis of ordinates
maps each point at the left side of the axis of ordinates to a point that lies in the same vertical line but jumps above its original position a number equal to its distance to the axis of ordinates
maps each point in the axis of ordinates to the same point.
(i) Find the subset V of the real plane whose inverse image under Ψ is U. Note that, when Ψ is restricted to V, this function is one to one and into U, continuous, its inverse is continuous, its derivatives of all orders exist, are continuous and the inverses of the derivatives are also continuous.
(ii) Assume f and g are real valued functions that are zero outside a compact subset of the real line and whose derivatives of all orders exist and are continuous.
Consider the function defined from the real plane to the real line that result from multiplying the composition of f with the projection over the first coordinate and the composition of g with the projection over the second coordinate.
Prove that integrating this function over U gives you the same result as if you compose this function with Ψ and integrate it over the subset of the real plane that lies at the left side of the vertical line that intersects the axis of abscissas at u.
(iii) Consider the same function defined in (ii) and assume g and f are both the Gaussian function.
Prove that its integral over U is equal to the square root of pi times the limit when b decreases without bounds of the integral from b to u over the square root of 2 of the Gaussian function.