What does this paradigm shift promise?

**Efficiency:**The current approach to understanding trends isn’t the most optimal, as we’ll uncover.**Computability:**Differentiable functions, although prevalent in calculus, are sparse in continuous spaces. A tool that captures trends without leaning heavily on differentiability would be groundbreaking.**Theoretical Foundation:**As trend applications mushroom, it’s incumbent upon us, the mathematicians, to lay a solid theoretical base.**Complementarity:**By championing a dedicated trend branch, we add a fresh dimension to calculus, bridging it with existing knowledge on rates.

Picture this: You’re in a car, trying to discern its movement direction. Do you first check the speed or glance at the gear indicator? Applying this logic to calculus, we ponder if it’s feasible to gauge local trends directly without the mediation of derivatives.

Through a series of mathematical transitions, we introduce a new trend operator. While the methodology might seem analogous to the derivative, there’s a distinct difference. While derivatives revolve around secant slopes turning into tangents, this trend operator gauges the trend of change in an interval encircling a point before applying the limit process.

This operator, christened as the “Detachment” operator, holds promise. It strips functions at their extremum points, yielding a step function that mirrors the original function’s monotonicity regions. You’ll find an elaborate post about such applications in the first part of this blog post.