91Ó°ÊÓ

Continuity is an essential concept in calculus that involves the behavior of functions without any interruptions or breaks in their graphs. For a function to be continuous at a point, the following conditions must be met:

• The function must be defined at the point.
• The function must have a limit as it approaches the point from both sides.
• The value of the function at the point must equal its limit as it approaches that point.

In the provided exercise, we are interested in whether the piecewise function \(g(x)\) is continuous at \(x = 1\). We analyze the behavior of \(g(x)\) by checking both segments of the piecewise definition: \(-1 \leq x \leq 1\) and \(x > 1\). At \(x = 1\), we calculate \(g(1)\) using its definition for \(-1 \leq x \leq 1\), and compare it to the value given by \(x - 10\) for values where \(x > 1\). For continuity, these values must match. By checking both sides, we confirm if the function exhibits any discontinuity at this point.

Differentiability is about the ability of a function to have a derivative at every point in its domain. A function must be continuous to be differentiable, but just being continuous is not enough for differentiability. The derivative essentially gives us the slope of the tangent line to the function at a particular point. To determine differentiability at a point, particularly at \(x = 1\) in our exercise, we examine the left-hand and right-hand derivatives of the function. To be differentiable at \(x = 1\), the derivatives approaching from both sides must be equal. Calculating limits of the derivative of \(g(x)\) as \(x\) approaches 1 from the left involves the interval \(-1 \leq x \leq 1\). For \(x > 1\), \(g(x)\) simplifies to a simple function \(x - 10\), where the derivative is straightforwardly 1. Any mismatch in these derivative values indicates that the function is not differentiable at the point \(x = 1\). This step helps determine if \(g(x)\) is smooth enough to not have any sharp corners or cusps at that point.

Piecewise functions are functions defined by multiple sub-functions, each applied to a certain interval of the domain. This can result in different behavioral characteristics within those intervals. In this problem, \(g(x)\) is a piecewise function:

• Within \(-1 \leq x \leq 1\), \(g(x)\) is defined by the minimum value of \(f(t)\) over the interval \([x, x+1]\).
• For \(x > 1\), \(g(x)\) switches to the linear function \(x - 10\).

Each segment can vary significantly, affecting both continuity and differentiability at transition points. Notably, the transition point at \(x = 1\) will require careful consideration, as it connects the two regimes of \(g(x)\). Calculating and comparing the sub-functions' values at this critical point allows us to check for continuity, while analyzing derivatives on either side informs us of differentiability. Understanding the nature of piecewise functions aids in visualizing and solving problems involving transitions between different function types within a single problem scope.