How to implement HandlerFunc without using DefaultServeMux

If I were to use the DefaultServeMux (which I designate by passing nil as the second argument to ListenAndServe), then I have access to http.HandleFunc, which you see used below in this example from the Go wiki:

func handler(w http.ResponseWriter, r *http.Request) {
    fmt.Fprintf(w, "Hi there, I love %s!", r.URL.Path[1:])
}

func main() {
    http.HandleFunc("/", handler)
    http.ListenAndServe(":8080", nil)
}

In my current code, I am not able to use the DefaultServeMux i.e. I'm passing a custom handler to ListenAndServe

    h := &mypackage.Handler{
        Database: mydb
    }
    http.ListenAndServe(":8080", h)

so I don't get the http.HandleFunc built in. However, I have to adapt some authorization code to my code base that requires something like http.HandleFunc. For example, if I had been using DefaultServeMux, when I hit the "/protected" route, I would want to go to the Protected handler, but only after passing through the h.AuthorizationHandlerFunc like this

   h.AuthorizationHandlerFunc(Protected)

However, since I'm not using DefaultServeMux, it's not working i.e. I'm not able to pass the Protected function (and have it called) to the AuthorizationHandlerFunc. This is the implementation of the AuthorizationHandlerFunc below. You can see below that Protected never gets called.

Question: how do I implement HandlerFunc in this situation (without using DefaultServeMux)?

func (h *Handler) AuthorizationHandlerFunc(next http.HandlerFunc) http.Handler{
     return http.HandlerFunc(func(w http.ResponseWriter, r *http.Request){
         h.AuthorizationMiddleWare(w, r, next)
   })
}

func (h *Handler) AuthorizationMiddleWare(w http.ResponseWriter, r *http.Request, next http.HandlerFunc){
     //other stuff happens
     log.Println("this is never getting called")
     next(w,r)
}
func (h *Handler)Protected(w http.ResponseWriter, r *http.Request){
     
      log.Println("this is never getting called")
}

Update
ServeHTTP is implemented on mypackage.Handler. Why is the Protected function not getting called, or, for that matter, the relevant code in the AuthorizationMiddleWare?

Re-implement your authorization middleware as a http.Handler :

type auth struct {
   DB *sql.DB
   UnauthorizedHandler http.Handler
}

func NewAuth(db *sql.DB, unauthorized http.Handler) *auth {
    return auth{db, unauthorized}
}

func (a *auth) Protected(h http.Handler) http.Handler {
    fn := func(w http.ResponseWriter, r *http.Request) {
        // Check whether the request is valid
        // If it's invalid, call your error func and make sure to *return* early!
        if !valid {
            a.UnauthorizedHandler.ServeHTTP(w, r)
            return
        }
        // Call the next handler on success
        h.ServeHTTP(w, r)
        return
    }

    return http.HandlerFunc(fn)
}

func someHandler(w http.ResponseWriter, r *http.Request) {
    io.WriteString(w, "Hello!\n")
}

func main() {
    auth := NewAuth(db, errorHandler)
    r := http.NewServeMux()
    // We have a http.Handler implementation that wraps a http.HandlerFunc
    // ... so we call r.Handle on our ServeMux and type-cast the wrapped func
    r.Handle("/protected", auth.Protected(http.HandlerFunc(someHandler)))
    // Just a simple http.HandlerFunc here
    r.HandleFunc("/public", someOtherHandler)

    log.Fatal(http.ListenAndServe(":8000", r))
}

Take a look at the httpauth lib I wrote for a different example with a ServeHTTP method. Both the above and explicitly creating a ServeHTTP method on your type are valid approaches.