Choose the Parent Armature Deform best suited to you.
Check your modifier panel settings. Know that your bones in your example and mine are perpendicular to the rather linear mesh portion example. For limb animation and rotation for organic models the bone is often inside the mesh for major limbs. Consider making the bones parallel to the mesh. Consider blurring the weights for gradual transitions.
Try these settings above when you parent the mesh to the armature.
You will achieve a more flowing gradient.
[With Automatic Weights]
Try the other settings.
You will achieve a more abrupt gradient.
[With Envelope Weights]
Below Closeup Abrupt Influence with a little curve.
Below Closeup Flowing Influence with a big curve.
Another way to get abrupt influence in my example is to choose Bone Envelopes only in the modifier panel.