We can also consider that investing with an emergency fund may come in a quick demand in seizing for an opportunity to invest,
Emergency funds are to be strictly kept for emergencies. It is best to maintain this discipline and not tamper with it unless we are faced with real emergencies since we do not know when the emergencies shows up and wouldn't want to be taken unawares and risk tampering our portfolio in order to solve them.
Emergency fund should not be use for anything except for an emergency and in urgent situation. If you mix the fund with the investing, you will loss your investment because you will use your investment to cover your emergency. That will make your investment amount reduce while you will difficult to accumulate more and more. We could be in an urgent situation anytime because no one can know the time. We don't have to tamper that fund because that fund is only for emergency funds and not for other things.
It is why we must have discipline and strictly to allocate the fund in the right place. If we are in the urgent situation, we can use that fund and cover it without disturb the other money in their places. We don't have to borrow other people money because we can handle our emergency need without a problem.