A Lewis acid catalyst composition comprising a specific mixed medium and a Lewis acid catalyst, wherein the Lewis acid catalyst is at least one compound selected from the group consisting of compounds respectively represented by the following formulae (1) and (2): [(R.sub.f.sup.1SO.sub.2)(R.sub.f.sup.2SO.sub.2)N].sub.nM, (1) and [(R.sub.f.sup.1SO.sub.2)(R.sub.f.sup.2SO.sub.2)(R.sub.f.sup.3SO.sub.2)C].- sub.nM. (2) A method for continuously performing a reaction which proceeds in the presence of the above-mentioned Lewis acid catalyst by using a specific mixed medium and the above-mentioned Lewis acid catalyst. A novel Lewis acid catalyst.