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Piecewise functions are defined by different expressions based on the input value, which makes them versatile for modeling situations where a process behaves differently in different scenarios. For instance, consider the function: \[f(x)=\begin{cases} x^{3} & \text{for} \ 0 \leq x < 1 \ x & \text{for} \ 1 \leq x \leq 2 \end{cases}\]Here, the function has two parts: different rules for different intervals of \(x\).
When dealing with piecewise functions, it's essential to check each segment where the rule changes, especially at the boundaries like \(x = 1\). This is crucial in analyzing continuity and differentiability, as each piece might behave differently.

Understanding limits is key to examining the behavior of functions as they approach certain points. For the given piecewise function:

• The left-hand limit \(\lim_{{x \to 1^-}} f(x)\) evaluates the function's behavior as \(x\) approaches from values less than 1.
• The right-hand limit \(\lim_{{x \to 1^+}} f(x)\) assesses the function as \(x\) comes from values greater than or equal to 1.

When both of these limits equal the actual function value at that point, the function is continuous. Calculations often involve substituting close values and simplifying. For our exercise, both limits equaled 1, matching \(f(1)\), ensuring the function's continuity at \(x = 1\).

Derivatives measure how a function changes. To determine differentiability at a specific point, we look at the rate of change from both sides of that point.
For the left-hand derivative approaching from values \(x < 1\): \[ \lim_{{h \to 0^-}} \frac{(1+h)^3 - 1}{h} = 3 \]This is calculated by expanding and simplifying the polynomial.
The right-hand derivative for values \(x \geq 1\): \[ \lim_{{h \to 0^+}} \frac{(1 + h) - 1}{h} = 1 \]This straightforward simplification shows the function's change from the right. If these derivatives differ, the function has a 'jump' or non-smooth transition, meaning it’s not differentiable at that point. In this exercise, the left-hand and right-hand derivatives were 3 and 1 respectively, hence, the function is not differentiable at \(x = 1\).

To determine if a function is continuous at a point, evaluate the function’s behavior as it approaches that point from both sides, as well as the actual value at the point.
The conditions for continuity at \(x = 1\) are:

• \( \lim_{x \to 1^-} f(x) \) must exist.
• \( \lim_{x \to 1^+} f(x) \) must exist.
• The limit evaluations must be equal, and equal to \(f(1)\).

If any condition is unmet, the function is not continuous at that point. For our piecewise function, these conditions were met: \[\lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1) = 1 \]Therefore, the function is continuous at \(x = 1\). Understanding these concepts helps in ensuring robust mathematical analysis and in identifying functions’ behavior accurately.