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Piecewise functions are defined by different expressions based on the input value. They can behave differently within various intervals of the domain. In this case, the function is defined as:

• \( g(x) = x + b \) when \( x < 0 \)
• \( g(x) = \cos x \) when \( x \geq 0 \)

To fully understand such functions, it's crucial to evaluate their behavior at the points where the definition switches. A point of particular interest is \( x = 0 \) as this is where the two expressions meet. Ensuring that there are no abrupt changes at these points is necessary for continuity and differentiability, which commonly affects how the function is defined or its parameters, like the value of \( b \).
This highlights the importance of carefully analyzing how each piece behaves in isolation and at boundary points.

Continuity at a point means the function doesn't "break" suddenly at that location. For this to happen, the limit of the function as it approaches the point from both directions must equal the function's value at that point.

Consider the function \( g(x) \) at \( x=0 \). The principal requirement for continuity here is:

• \( \lim_{x \to 0^-}(x+b) = g(0) = \cos(0) \)

Since the limit from the left-hand side, \( \lim_{x \to 0^-}(x+b) \), equals \( b \) and the function's value at \( x=0 \), \( \cos(0) \), is 1, \( b \) must be 1 for the function to be continuous.

Thus, understanding continuity involves ensuring left and right-hand limits agree with the functional value, crucial for evaluating piecewise functions at junction points.

For differentiability at a point, a function needs to have a defined tangent or simply a smooth graph at that point.
Mathematically, this means:

• The derivative exists at that point.

For the given piecewise function \( g(x) \), check the derivatives approaching \( x=0 \) from either side. For \( x < 0 \), the derivative of \( x + b \) is 1. For \( x \geq 0 \), the derivative of \( \cos x \) is \(-\sin(x)\), specifically \( \sin(0) = 0 \) right at \( x = 0 \).
Since these derivatives don't match (1 is not equal to 0), the function is not differentiable at \( x=0 \). Differentiability requires not just continuity but also consistent slopes, highlighting the importance of both conditions.

Understanding left-hand and right-hand limits is crucial for analyzing piecewise functions around points where they change forms. These limits evaluate the function's behavior as it approaches a point from the left side or the right side separately.

Here's how:

• Left-hand limit, \( \lim_{x \to a^-} f(x) \), looks at how \( f(x) \) behaves as \( x \) approaches \( a \) from values less than \( a \).
• Right-hand limit, \( \lim_{x \to a^+} f(x) \), analyzes \( f(x) \) as \( x \) approaches \( a \) from values greater than \( a \).

To ensure continuity, these two must not only exist but be equal at the function's turning point, equating to the function's actual value. For differentiability, this equality is necessary at the derivative level.

In our example, while left and right-hand limits could equate to achieve continuity, the left and right-hand derivatives do not, which is why the function isn’t differentiable at \( x = 0 \).