@Ramin, I tried to recreate a scenario similar to yours but could not repro the issue. Here are the steps:
Setup an Event Hub-triggered Function (using latest version 0.7 as of 10/27/2016)
Setup a simple console app sender to my Event Hub that sends 10 messages with time stamps and message ids. Each message is sent 6 minutes apart. Here's the output for my console app that logs the sent messages,
Sent message: Timestamp: 10/27/2016 2:27:06 PM, MessageId: 1
Sent message: Timestamp: 10/27/2016 2:33:07 PM, MessageId: 2
Sent message: Timestamp: 10/27/2016 2:39:07 PM, MessageId: 3
Sent message: Timestamp: 10/27/2016 2:45:07 PM, MessageId: 4
Sent message: Timestamp: 10/27/2016 2:51:08 PM, MessageId: 5
Sent message: Timestamp: 10/27/2016 2:57:08 PM, MessageId: 6
Sent message: Timestamp: 10/27/2016 3:03:08 PM, MessageId: 7
Sent message: Timestamp: 10/27/2016 3:09:08 PM, MessageId: 8
Sent message: Timestamp: 10/27/2016 3:15:08 PM, MessageId: 9
Sent message: Timestamp: 10/27/2016 3:21:08 PM, MessageId: 10
Stopped sending messages.
Here are the log entries for my Function,
2016-10-27T21:27:26.516 Function started (Id=d5ed0e23-2b0e-4e0b-a858-f1e497dcac8c)
2016-10-27T21:27:26.516 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 2:27:06 PM, MessageId: 1
2016-10-27T21:27:26.516 Function completed (Success, Id=d5ed0e23-2b0e-4e0b-a858-f1e497dcac8c)
2016-10-27T21:33:26.667 Function started (Id=9525a046-86fb-4499-9d4f-b0d0fd0d0829)
2016-10-27T21:33:26.667 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 2:33:07 PM, MessageId: 2
2016-10-27T21:33:26.667 Function completed (Success, Id=9525a046-86fb-4499-9d4f-b0d0fd0d0829)
2016-10-27T21:39:42.074 Function started (Id=e2d528c9-f1b9-41aa-9c38-669c57c8182a)
2016-10-27T21:39:42.074 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 2:39:07 PM, MessageId: 3
2016-10-27T21:39:42.074 Function completed (Success, Id=e2d528c9-f1b9-41aa-9c38-669c57c8182a)
2016-10-27T21:45:26.794 Function started (Id=ff5325af-5dab-48fb-95b1-8318fada3c8c)
2016-10-27T21:45:26.794 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 2:45:07 PM, MessageId: 4
2016-10-27T21:45:26.794 Function completed (Success, Id=ff5325af-5dab-48fb-95b1-8318fada3c8c)
2016-10-27T21:51:26.875 Function started (Id=01d3fbf9-b772-4076-8fbf-783dc16677a7)
2016-10-27T21:51:26.875 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 2:51:08 PM, MessageId: 5
2016-10-27T21:51:26.875 Function completed (Success, Id=01d3fbf9-b772-4076-8fbf-783dc16677a7)
2016-10-27T21:57:26.989 Function started (Id=443d96e6-af97-460a-9f70-9dbc2eaf2f37)
2016-10-27T21:57:26.989 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 2:57:08 PM, MessageId: 6
2016-10-27T21:57:26.989 Function completed (Success, Id=443d96e6-af97-460a-9f70-9dbc2eaf2f37)
2016-10-27T22:03:27.088 Function started (Id=9eefe03d-8e78-4119-a453-96655ea01824)
2016-10-27T22:03:27.088 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 3:03:08 PM, MessageId: 7
2016-10-27T22:03:27.088 Function completed (Success, Id=9eefe03d-8e78-4119-a453-96655ea01824)
2016-10-27T22:10:01.872 Function started (Id=34c9b277-059d-4839-9dce-aeb03afb2871)
2016-10-27T22:10:01.872 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 3:09:08 PM, MessageId: 8
2016-10-27T22:10:01.872 Function completed (Success, Id=34c9b277-059d-4839-9dce-aeb03afb2871)
2016-10-27T22:15:27.706 Function started (Id=50eda659-c68f-4e61-a942-32160668fd5c)
2016-10-27T22:15:27.706 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 3:15:08 PM, MessageId: 9
2016-10-27T22:15:27.706 Function completed (Success, Id=50eda659-c68f-4e61-a942-32160668fd5c)
2016-10-27T22:21:52.162 Function started (Id=fa8f8059-013f-42f9-8047-391d4e3fb4a3)
2016-10-27T22:21:52.162 C# Event Hub trigger function processed a message: Timestamp: 10/27/2016 3:21:08 PM, MessageId: 10
2016-10-27T22:21:52.162 Function completed (Success, Id=fa8f8059-013f-42f9-8047-391d4e3fb4a3)
As you can see, all the messages arrived without duplication. Can you confirm that which each invocation of your Function (you can check your Function logs), that it actually ran to completion successfully?
If you see the following message,
Function completed (Success, Id=<some Guid>)
for every invocation of your Function, then the message(s) for that execution should have been check-pointed, and you should not be processing the message(s) again in the next invocation.
However, if you do not see the message or see an error message, then the Function run failed, which will result in the message(s) not being check-pointed. When the Function gets triggered again, it will pick up from the last checkpoint, which could be the first message, if your Function never succeeded in processing that message.