diff --git a/optax/transforms/_clipping.py b/optax/transforms/_clipping.py
index 824998e53..95cefbf27 100644
--- a/optax/transforms/_clipping.py
+++ b/optax/transforms/_clipping.py
@@ -99,7 +99,11 @@ def update_fn(updates, state, params=None):
     # once analyzed how it affects backprop through update (e.g. meta-gradients)
     # g_norm = jnp.maximum(max_norm, g_norm)
     # updates = jax.tree.map(lambda t: (t / g_norm) * max_norm, updates)
-    trigger = jnp.squeeze(g_norm < max_norm)
+    # Non-strict comparison: clipping updates to their own norm is the identity,
+    # and taking the pass-through branch at equality avoids computing 0/0 = NaN
+    # when the global norm and max_norm are both zero (e.g. all-zero gradients
+    # under a schedule-driven max_norm that reaches zero).
+    trigger = jnp.squeeze(g_norm <= max_norm)
     utils.check_rank(trigger, 0)  # A scalar.
 
     def clip_fn(t):
diff --git a/optax/transforms/_clipping_test.py b/optax/transforms/_clipping_test.py
index d7fcdb45e..4b2f03879 100644
--- a/optax/transforms/_clipping_test.py
+++ b/optax/transforms/_clipping_test.py
@@ -75,6 +75,23 @@ def test_clip_by_global_norm(self):
       updates_step, _ = clipper.update(self.per_step_updates, None)
       test_utils.assert_trees_all_close(updates, updates_step)
 
+  def test_clip_by_global_norm_zero_norm_zero_max_norm(self):
+    # 0 / 0 in the clip branch used to produce NaN updates when the global
+    # norm and max_norm were both zero (e.g. all-zero gradients under a
+    # schedule-driven max_norm that reaches zero).
+    clipper = _clipping.clip_by_global_norm(0.0)
+    zero_updates = jax.tree.map(jnp.zeros_like, self.per_step_updates)
+    updates, _ = clipper.update(zero_updates, None)
+    test_utils.assert_trees_all_close(updates, zero_updates)
+
+  def test_clip_by_global_norm_at_equality_is_identity(self):
+    # Clipping updates to exactly their own norm is a no-op: pins the
+    # boundary semantics of the g_norm <= max_norm comparison.
+    g_norm = optax.tree.norm(self.per_step_updates)
+    clipper = _clipping.clip_by_global_norm(g_norm)
+    updates, _ = clipper.update(self.per_step_updates, None)
+    test_utils.assert_trees_all_close(updates, self.per_step_updates)
+
   def test_adaptive_grad_clip_with_axis(self):
     """Test adaptive_grad_clip with custom axis parameter."""