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Before diving into differentiability and other topics, grasping continuity is crucial. A function is continuous at a specific point if there are no "jumps" or "breaks" in the graph at that point. To be mathematically precise, a function \( f(x) \) is continuous at a point \( x = c \) if the following conditions are met:

• The function \( f(c) \) is defined.
• The limit \( \lim_{{x \to c}} f(x) \) exists.
• \( \lim_{{x \to c}} f(x) = f(c) \)

In the case of our function \( f(x) = \begin{cases} e^x, & x \leq 0 \ |x - 1|, & x > 0 \end{cases} \), we first checked continuity at \( x = 0 \). By calculating \( \lim_{{x \to 0^-}} f(x) = 1 \), \( \lim_{{x \to 0^+}} f(x) = 1 \), and knowing \( f(0) = 1 \), we see the function is continuous at this point. However, it's not continuous at \( x = 1 \) because the limit from the left, \( e \), is different from the limit from the right, which is \( 0 \). Understanding continuity helps in laying a foundation for the graphical behavior of functions.

While continuity is about the absence of jumps, differentiability is about the smoothness of the curve. For a function to be differentiable at a point \( x = c \), it needs to have a defined slope or tangent line at that point. Essentially, this means that both the left-hand and right-hand derivatives at \( x = c \) must exist and be equal. In formal terms:

• The left-hand derivative \( \lim_{{h \to 0^-}} \frac{f(c+h) - f(c)}{h} \) exists.
• The right-hand derivative \( \lim_{{h \to 0^+}} \frac{f(c+h) - f(c)}{h} \) exists.
• Both derivatives are equal: \( f'(c^-) = f'(c^+) \)

In our exercise, checking differentiability at \( x = 0 \) shows two different derivatives: \( f'(0^-) = 1 \) and \( f'(0^+) = -1 \). Since they aren't equal, the function isn't differentiable at \( x = 0 \). Similarly, at \( x = 1 \), the left-hand derivative \( e \) does not match the right-hand derivative \( 1 \), so it's not differentiable there either. This reveals that even if a function is continuous at a point, it may not be differentiable there, highlighting the subtleties between these two concepts.

Piecewise functions are compositions of multiple sub-functions, each applying to different parts of the function's domain. They can often be spotted by their unique representation, where each sub-function is applied for certain intervals. Examining how each piece behaves individually can help determine properties like continuity and differentiability across the entire function.
With our example function, \( f(x) = \begin{cases} e^x, & x \leq 0 \ |x - 1|, & x > 0 \end{cases} \), there are two distinct behaviors based on the interval of \( x \). For \( x \leq 0 \), the function is \( e^x \), which is smooth and has well-defined derivatives, indicating that it's both continuous and differentiable on its interval. For \( x > 0 \), we have \( |x - 1| \), which changes description based on whether \( x \) is less than or greater than 1.

• Before the point \( x = 1 \), \( |x - 1| \) behaves like \( 1 - x \).
• After \( x \), it behaves like \( x - 1 \) in terms of slope.

Handling such functions requires attention to these individual pieces, where their specific rules determine the overall function's properties at transition points. Exploring piecewise functions helps develop a fuller understanding of how functions can behave in sections and underlines the importance of evaluating each piece on its own merits to discern the overall behavior at critical points.