Mitering the vertices produces chamfered edges that meet at an edge at the corners.
This example will use the following model.

Make sure Round vertex is disabled.
Select the object whose edges are to be chamfered. Each edge is marked with a blue square.
Set the chamfer offset values in the Inspector Bar, and select the edge to chamfer.
The edge is chamfered. To chamfer another edge, select the object again and select the next edge.
Continue to select edges to chamfer.

Any common corners will meet at a sharp edge.

Pick Smooth Sequence

Adjacent edges are smoothly connected if they are connected by an arc or rounded vertex. When working with Round vertex, smoothly connected edges are automatically selected. But this is not the case when Round vertex is turned off.

Start with a box with one filleted edge, using a large radius.
Activate Chamfer without Round vertex, and chamfer one edge, using a smaller offset.

This is the result - only the selected edge was chamfered.
Undo, and this time select Pick smooth sequence.
This time when you select one of these edges for chamfering, all edges in the smooth chain are selected.

This is the result - all edges in the chain are chamfered.

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