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Differentiability is a crucial concept in calculus that refers to the existence of a derivative for a function at a particular point. In simpler terms, a function is differentiable at a point if it has a specific slope at that point. A piecewise function is a function composed of several sub-functions, each applicable to a certain interval. For a piecewise- C^{1} function, differentiability means having a continuous and continuously differentiable derivative except at some points where it may switch from one piece to another.
The idea of proving differentiability is about verifying if a function transitions smoothly from one part to another without abrupt changes. At any point where a function is not differentiable, there could be a sharp turn or corner. However, within the intervals of differentiability, the presence of a continuous derivative ensures smoothness and the ability to determine the function's slope.

• If a derivative exists and is continuous in an open interval around a point, except possibly at that point itself, the function is called piecewise- C^{1} at that point.
• Even if a function is not differentiable at a point, it can still be continuous, meaning there is no sudden jump in the function's value.
• The absence of differentiability often indicates a corner or cusp at that point.

One-sided derivatives help us understand the behavior of functions at points where they are not differentiable. Unlike regular derivatives, which consider changes from both sides of a point, one-sided derivatives focus on the rate of change approaching from one specific direction.

To define one-sided derivatives for a functionf at a pointc:

• The left-hand derivative,\( f'_{-}(c) \) is expressed as \[ \lim_{h \to 0^{-}} \frac{f(c+h) - f(c)}{h}, \]
• The right-hand derivative,\( f'_{+}(c) \) is given by \[ \lim_{h \to 0^{+}} \frac{f(c+h) - f(c)}{h}. \]

For piecewise-C^{1} functions, these one-sided derivatives exist at points of non-differentiability because the function is continuously differentiable in the surrounding intervals. Thus, although the regular derivative might not exist atc, the one-sided derivatives can. This makes them invaluable for analyzing the behavior at transition points where the function changes its piece.

Continuity is the property of a function where its values change in a smooth way, without jumps or gaps. A function is said to be continuous at a point if the limit as we approach that point is equal to the function's value at that point.
For piecewise functions, ensuring continuity involves checking that each sub-function meets at the boundaries without a sudden discrepancy. Even if a function is not differentiable at a point, it can still be continuous. Continuity ensures the function’s graph is connected, but does not guarantee smoothness like differentiability does.

• A function is continuous at a point\(c\) if\( \lim_{x \to c} f(x) = f(c) \)
• Continuity across boundaries of piecewise functions means the value from the left matches the value from the right at every breakpoint.
• For a piecewise-C^{1} function, continuity implies a seamless joining of the sub-functions.

Understanding these properties allows mathematicians to precisely describe and work with functions that may not be smooth everywhere, yet remain predictable and consistent in behavior.