We can use the Kudu WebJob API to start or stop a continuous job
POST https://{user}:{password}@{sitename}.scm.azurewebsites.net/api/continuouswebjobs/{job name}/start
We can get user and password info from the Azure WebApp publishSetting file. And We can download the file from the portal

Test the Rest API using the fiddler

Note: How to create Azure function please refer to the document.
Update:
is there a way I can trigger a specific function in that continuous job?
Based on my experience, I can't find a way or Rest API to trigger a sepcific function in continuous job directly.
My work around is that we can use Webjob QueueTrigger. According the queue message info to trigger the sepcific function.
We can create a WebJob with QueueTrigger
The following is my detail steps according to your code.
1.Create an Azure storage queue for trigger

2.Create a Webjob project and add the following code
public static void SyncData([QueueTrigger("backup")] string logMessage, TextWriter logger)
{
Console.WriteLine($"Start time:{DateTime.Now}");
switch (logMessage.ToLower())
{
case "syncall":
SyncAll(logger);
break;
case "syncbranches":
SyncBranches(logger);
break;
case "synccustomers":
SyncCustomers(logger);
break;
case "syncinventory":
SyncInventory(logger);
break;
default:
Console.WriteLine("Default case");
break;
}
Console.Write($"Endtime:{DateTime.Now}");
}
[NoAutomaticTrigger]
public static Task SyncAll(TextWriter log)
{
Console.WriteLine("SyncAll :"+DateTime.Now);
return null;
//await Task.Delay(10);
}
[NoAutomaticTrigger]
public static Task SyncBranches(TextWriter log)
{
Console.WriteLine("SyncBranches :" + DateTime.Now);
return null;
}
[NoAutomaticTrigger]
public static Task SyncCustomers(TextWriter log)
{
Console.WriteLine("SyncCustomers :" + DateTime.Now);
return null;
}
[NoAutomaticTrigger]
public static Task SyncInventory(TextWriter log)
{
Console.WriteLine("SyncInventory :" + DateTime.Now);
return null;
}
3.Use Azure Storage Queue REST API to create a queue message.
4.Check result the from the console.
