  We describe a method for approximating a single-variable function $f$ using
persistence diagrams of sublevel sets of $f$ from height functions in different
directions. We provide algorithms for the piecewise linear case and for the
smooth case. Three directions suffice to locate all local maxima and minima of
a piecewise linear continuous function from its collection of directional
persistence diagrams, while five directions are needed in the case of smooth
functions with non-degenerate critical points.
  Our approximation of functions by means of persistence diagrams is motivated
by a study of importance attribution in machine learning, where one seeks to
reduce the number of critical points of signal functions without a significant
loss of information for a neural network classifier.
