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Discontinuous differential quotients appear for example in mechanics in the case of touching of two bodies, if pecise modelling of the transient behaviour is not of interest.

Example: Roll Pendulum

the rotation energy is taken into account, but the reflection, that is a jumping of the ball on the bed, is not considered.


DIFFERENTIAL EQUATION
s' := v;
v' := F / (m + J/(r*r));
END

IF    s < -c
DO F := m*g*SIN(alpha);  END
ELSIF s >  c
DO F := -m*g*SIN(alpha); END
ELSE
DO F := 0;               END

IF    s < -c
DO x := -c-(c-s)*COS(alpha); END
ELSIF s >  c
DO x :=  c+(s-c)*COS(alpha); END
ELSE
DO x := s;                   END
Roll Pendulum

A positive or negative propelling force acts on the ball along the inclined plane, while the horizontal center part shows no such force. At the transition between planes the force therefore abruptly changes its magnitude, a discontinuous course.

Comment: Both case distinctions can be collected syntactically into a single one.

Differential quotients with discontinuities cause great errors in integration that do not decrease by smaller step sizes. That led to special provisions that asure a consistent behaviour of integrations.