For those only in need of single servicebus client a simple singleton would suffice

(Tested with .Net 6 Azure Functions v4):

using Azure.Messaging.ServiceBus;
using Microsoft.Azure.Functions.Extensions.DependencyInjection;
using Microsoft.Extensions.DependencyInjection;
using System;

[assembly: FunctionsStartup(typeof(YourProjName.Startup))]

namespace YourProjName
{
 public class Startup : FunctionsStartup
 {
    public override void Configure(IFunctionsHostBuilder builder)
    {
       
        var serviceBusConnectionString = Environment.GetEnvironmentVariable("ServiceBusConnectionString");
        if (string.IsNullOrEmpty(serviceBusConnectionString))
        {
            throw new InvalidOperationException(
                "Please specify a valid ServiceBusConnectionString in the Azure Functions Settings or your local.settings.json file.");
        }

        //using AMQP as transport
        builder.Services.AddSingleton((s) => {
            return new ServiceBusClient(serviceBusConnectionString, new ServiceBusClientOptions() { TransportType = ServiceBusTransportType.AmqpWebSockets }); 
        });

    }
 }
}

Then you could use it in the Azure Function constructor:

public Function1(ServiceBusClient client)
   {
       _client = client;
   }

ServiceBusClientBuilderExtensions.AddServiceBusClient is an extension method of IAzureClientFactoryBuilder:

public static IAzureClientBuilder<ServiceBusClient, ServiceBusClientOptions> AddServiceBusClient<TBuilder>(this TBuilder builder, string connectionString)
            where TBuilder : IAzureClientFactoryBuilder

To get an instance of IAzureClientFactoryBuilder, you need to call AzureClientServiceCollectionExtensions.AddAzureClients(IServiceCollection, Action<AzureClientFactoryBuilder>) for a given IServiceCollection, which provides a delegate giving an instance of IAzureClientFactoryBuilder. (this method is in the Microsoft.Extensions.Azure NuGet package)

To call that method, you can use the IServiceCollection provided by IFunctionsHostBuilder. With all of that, what you have should look something like:

public override void Configure(IFunctionsHostBuilder builder)
{
    builder.Services.AddAzureClients(clientsBuilder =>
    {
        clientsBuilder.AddServiceBusClient(Typsy.Domain.Configuration.Settings.Instance().Connections.ServiceBusPrimary)
          // (Optional) Provide name for instance to retrieve by with DI
          .WithName("Client1Name")
          // (Optional) Override ServiceBusClientOptions (e.g. change retry settings)
          .ConfigureOptions(options =>
          {
              options.RetryOptions.Delay = TimeSpan.FromMilliseconds(50);
              options.RetryOptions.MaxDelay = TimeSpan.FromSeconds(5);
              options.RetryOptions.MaxRetries = 3;
          });
    });
}

To retrieve the named instance, instead of using ServiceBusClient as the injected type you use IAzureClientFactory<ServiceBusClient>. The ServiceBusClient is a Singleton regardless of whether you use a named instance or not.

 public Constructor(IAzureClientFactory<ServiceBusClient> serviceBusClientFactory)
 {
     // Wherever you need the ServiceBusClient
     ServiceBusClient singletonClient1 = serviceBusClientFactory.CreateClient("Client1Name")
 }