I have a morning routine which "check email for emergencies" is on the list, but I do not do that until I sit at my desk. And, checking email is a calendared item. Meaning, some days it happens at 8am, others it might not be until in the afternoon; in particular if my exercise is calendared for the morning that day.
What does this do for my clients? Well, my clients rarely expect an immediate response and the reality is most are good if I respond within 24-48 hours. In the event of an emergency, they can utilize sms and it gives me the option of determining which quadrant of "tasks" it falls into.