  The introduction of structural defects in otherwise periodic media is well
known to grant exceptional space control and localization of waves in various
physical fields, including elasticity.
  Despite the variety of designs proposed so far, most of the approaches derive
from contextual modifications that do not translate into a design paradigm due
to the lack of a general theory. Few exceptions include designs endowed with
topological dispersion bands, which, however, require changes over substantial
portions of the structure.
  To overcome these limitations, here we introduce a new rationale based on
real-space topology to achieve localized modes in continuous elastic media. We
theoretically predict and experimentally demonstrate the spectral flow of a
localized mode across a bulk frequency gap by modulating a single structural
parameter at any chosen location in the structure.
  The simplicity and generality of this approach opens new avenues in designing
wave-based devices for energy localization and control.
