diff --git a/llvm/lib/Transforms/IPO/DeadArgumentElimination.cpp b/llvm/lib/Transforms/IPO/DeadArgumentElimination.cpp
--- a/llvm/lib/Transforms/IPO/DeadArgumentElimination.cpp
+++ b/llvm/lib/Transforms/IPO/DeadArgumentElimination.cpp
@@ -268,9 +268,12 @@
   if (!Fn.hasExactDefinition())
     return false;
 
-  // Functions with local linkage should already have been handled, except the
-  // fragile (variadic) ones which we can improve here.
-  if (Fn.hasLocalLinkage() && !Fn.getFunctionType()->isVarArg())
+  // Functions with local linkage should already have been handled, except if
+  // they are fully alive (e.g., called indirectly) and except for the fragile
+  // (variadic) ones. In these cases, we may still be able to improve their
+  // statically known call sites.
+  if ((Fn.hasLocalLinkage() && !LiveFunctions.count(&Fn)) &&
+      !Fn.getFunctionType()->isVarArg())
     return false;
 
   // Don't touch naked functions. The assembly might be using an argument, or
diff --git a/llvm/test/Transforms/DeadArgElim/fct_ptr.ll b/llvm/test/Transforms/DeadArgElim/fct_ptr.ll
new file mode 100644
--- /dev/null
+++ b/llvm/test/Transforms/DeadArgElim/fct_ptr.ll
@@ -0,0 +1,67 @@
+; RUN: opt -S %s -deadargelim -o - | FileCheck %s
+; In that test @internal_fct is used by an instruction
+; we don't know how to rewrite (the comparison that produces
+; %cmp1).
+; Because of that use, we used to bail out on removing the
+; unused arguments for this function.
+; Yet, we should still be able to rewrite the direct calls that are
+; statically known, by replacing the related arguments with undef.
+; This is what we check on the call that produces %res2.
+
+define i32 @call_indirect(i32 (i32, i32, i32)* readnone %fct_ptr, i32 %arg1, i32 %arg2, i32 %arg3) {
+; CHECK-LABEL: @call_indirect(
+; CHECK-NEXT:    [[CMP0:%.*]] = icmp eq i32 (i32, i32, i32)* [[FCT_PTR:%.*]], @external_fct
+; CHECK-NEXT:    br i1 [[CMP0]], label [[CALL_EXT:%.*]], label [[CHK2:%.*]]
+; CHECK:       call_ext:
+; CHECK-NEXT:    [[RES1:%.*]] = tail call i32 @external_fct(i32 undef, i32 [[ARG2:%.*]], i32 undef)
+; CHECK-NEXT:    br label [[END:%.*]]
+; CHECK:       chk2:
+; CHECK-NEXT:    [[CMP1:%.*]] = icmp eq i32 (i32, i32, i32)* [[FCT_PTR]], @internal_fct
+; CHECK-NEXT:    br i1 [[CMP1]], label [[CALL_INT:%.*]], label [[CALL_OTHER:%.*]]
+; CHECK:       call_int:
+; CHECK-NEXT:    [[RES2:%.*]] = tail call i32 @internal_fct(i32 undef, i32 [[ARG2]], i32 undef)
+; CHECK-NEXT:    br label [[END]]
+; CHECK:       call_other:
+; CHECK-NEXT:    [[RES3:%.*]] = tail call i32 @other_fct(i32 [[ARG2]])
+; CHECK-NEXT:    br label [[END]]
+; CHECK:       end:
+; CHECK-NEXT:    [[FINAL_RES:%.*]] = phi i32 [ [[RES1]], [[CALL_EXT]] ], [ [[RES2]], [[CALL_INT]] ], [ [[RES3]], [[CALL_OTHER]] ]
+; CHECK-NEXT:    ret i32 [[FINAL_RES]]
+;
+  %cmp0 = icmp eq i32 (i32, i32, i32)* %fct_ptr, @external_fct
+  br i1 %cmp0, label %call_ext, label %chk2
+
+call_ext:
+  %res1 = tail call i32 @external_fct(i32 %arg1, i32 %arg2, i32 %arg3)
+  br label %end
+
+chk2:
+  %cmp1 = icmp eq i32 (i32, i32, i32)* %fct_ptr, @internal_fct
+  br i1 %cmp1, label %call_int, label %call_other
+
+call_int:
+  %res2 = tail call i32 @internal_fct(i32 %arg1, i32 %arg2, i32 %arg3)
+  br label %end
+
+call_other:
+  %res3 = tail call i32 @other_fct(i32 %arg1, i32 %arg2, i32 %arg3)
+  br label %end
+
+end:
+  %final_res = phi i32 [%res1, %call_ext], [%res2, %call_int], [%res3, %call_other]
+  ret i32 %final_res
+}
+
+
+define i32 @external_fct(i32 %unused_arg1, i32 %arg2, i32 %unused_arg3) {
+  ret i32 %arg2
+}
+
+define internal i32 @internal_fct(i32 %unused_arg1, i32 %arg2, i32 %unused_arg3) {
+  ret i32 %arg2
+}
+
+define internal i32 @other_fct(i32 %unused_arg1, i32 %arg2, i32 %unused_arg3) {
+  ret i32 %arg2
+}
+