main.go

package main

import (
	"fmt"
	"net/http"

	"github.com/apex/go-apex"
	"github.com/apex/go-apex/proxy"
)

func main() {
	h := http.NewServeMux()
	h.HandleFunc("/hello", hello)
	h.HandleFunc("/world", world)
	apex.Handle(proxy.Serve(h))
}

func hello(w http.ResponseWriter, r *http.Request) {
	fmt.Fprintf(w, "Hello")
}

func world(w http.ResponseWriter, r *http.Request) {
	fmt.Fprintf(w, "World")
}

https://github.com/apex/go-apex

Lets you run regular net/http servers on AWS API Gateway via https://github.com/apex/apex with just apex.Handle(proxy.Serve(h)), or apex.Handle(proxy.Serve(nil)) for DefaultServeMux, otherwise everything else is normal.