How to let a contact bypass iOS Do Not Disturb mode.
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The iPhone’s Do Not Disturb feature is a great way to make sure your phone doesn’t disturb you when you’re sleeping or busy. But if you want a certain contact to be able to contact you, you can whitelist them. That is how.
How the iOS Do Not Disturb feature works
Do Not Disturb automatically silences all calls, notifications, and alerts you receive when your iPhone is locked. The feature can be enabled by tapping on the “Crescent Moon” icon in Control Center.

If you touch 3D Touch / Haptic Touch on the Do Not Disturb icon, you can enable it from one hour or until you leave your current location. You can also set Do Not Disturb on a schedule by going to Settings > Do Not Disturb > Schedule.
By default, Do Not Disturb works like an all or nothing switch. However, you can add exceptions to the Do Not Disturb filter in several ways.

How to whitelist favorites in Do Not Disturb
The Favorites tab in the Phone app is a modern take on speed dialing. Add a couple of numbers you call frequently for a one-touch connection. The Favorites feature also has a couple of other benefits, like the option to add the Favorites widget to your lock screen.
The favorites list also integrates directly with the Do Not Disturb feature. To set things up, open the Settings app, go to “Do Not Disturb,” select “Phone,” and then tap “Allow Calls From.”

Next, change the settings to “Favorites”. With this set, anyone who calls you from your favorites list will be able to join.
iOS also has an option in the Do Not Disturb section called “Repeated Calls”. When you enable this feature, a second call from the same person will be let through within three minutes.

How to add contacts to favorites
If you haven’t set up the Favorites list, you can do so by going to the Phone app and tapping on the “Favorites” tab. Add your family members and close friends to this list by tapping on the “More” button in the top left corner.
Now search for a contact from your contact book.
Tap on the contact and you’ll see a pop-up with all of your available contact information (including alternate numbers and FaceTime options).

Choose the “Call” option and then select your number.
They will be added to your favorites list. Repeat the process for all the contacts you want to whitelist for Do Not Disturb.
Allow contacts through using emergency bypass
The Favorites method to allow calls through will not work for everyone. I, for example, use Favorites as a shortcut to access my frequently used contacts, which includes all my family members and friends.

In an emergency, I want only a few contacts to go through the Do Not Disturb filter. If you’re in the same boat, you can use the emergency bypass feature added in iOS 10.
The emergency bypass feature works on a per-contact basis and includes controls for calls and text messages. Once you enable emergency bypass for a contact, your call or text message can come through even if you have Do Not Disturb enabled and your phone is on silent mode.
To enable emergency forwarding for a contact, open the Phone app and search for the contact.
Touch the contact to open their contact card. Then, tap on the “Edit” button at the top right corner.

Swipe down until you find the “Ringtone” option and tap on it.
From the top of this screen, tap the toggle next to “Emergency Bypass” to enable the feature. Tap “Done” to save the settings and return to the previous screen.
You can do the same with text messages by tapping on “Text Tone” and enabling the “Emergency Bypass” feature on the next screen.

You can repeat this process for all the contacts you want to let through the Do Not Disturb filter. Once it’s set up, whenever they call you or send you a message, your iPhone will ring.