Understanding the Given Piecewise Function

We are given a function \(f(x)\) defined in different ways over different intervals. Such a function is called a piecewise function. We need to determine if this function is a continuous function at the points where its definition changes, which are \(x = 0\) and \(x = 1\).

A function is continuous at a point \(a\) if the following three conditions are met:

1. The function is defined at \(a\), i.e., \(f(a)\) exists.
2. The limit of the function exists as \(x\) approaches \(a\), i.e., \(\lim_{x \to a} f(x)\) exists. This means the left-hand limit (LHL) and the right-hand limit (RHL) at \(a\) are equal. Mathematically, \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)\).
3. The limit of the function at \(a\) is equal to the function's value at \(a\), i.e., \(\lim_{x \to a} f(x) = f(a)\).

Let's check the continuity at the specified points for our given piecewise function:

The function is defined as:

• \(f(x) = -x\), when \(x \le 0\)
• \(f(x) = x\), when \(0 < x < 1\)
• \(f(x) = 2 - x\), when \(x \ge 1\)

Checking Continuity at x = 0

We will check the three conditions for continuity at \(x = 0\).

1. Function value at x = 0: For \(x \le 0\), \(f(x) = -x\). So, \(f(0) = -(0) = 0\). The function is defined at \(x = 0\).
2. Limit at x = 0: We need to find the left-hand limit and the right-hand limit at \(x = 0\).
   • Left-hand limit (LHL): As \(x \to 0^-\), \(x < 0\). For \(x < 0\), \(f(x) = -x\).

     \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = -(0) = 0\)

   • Right-hand limit (RHL): As \(x \to 0^+\), \(x > 0\). For \(0 < x < 1\), \(f(x) = x\).

     \(\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x) = 0\)

   Since the LHL (\(0\)) equals the RHL (\(0\)), the limit \(\lim_{x \to 0} f(x)\) exists and is equal to \(0\).

3. Comparing function value and limit: We have \(f(0) = 0\) and \(\lim_{x \to 0} f(x) = 0\).

   Since \(\lim_{x \to 0} f(x) = f(0)\), the function \(f(x)\) is continuous at \(x = 0\). This confirms the first part of evaluating the continuous function.

Checking Continuity at x = 1

Now, let's check the three conditions for continuity at \(x = 1\).

1. Function value at x = 1: For \(x \ge 1\), \(f(x) = 2 - x\). So, \(f(1) = 2 - 1 = 1\). The function is defined at \(x = 1\).
2. Limit at x = 1: We need to find the left-hand limit and the right-hand limit at \(x = 1\).
   • Left-hand limit (LHL): As \(x \to 1^-\), \(x < 1\). For \(0 < x < 1\), \(f(x) = x\).

     \(\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x) = 1\)

   • Right-hand limit (RHL): As \(x \to 1^+\), \(x > 1\). For \(x \ge 1\), \(f(x) = 2 - x\).

     \(\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2 - x) = 2 - 1 = 1\)

   Since the LHL (\(1\)) equals the RHL (\(1\)), the limit \(\lim_{x \to 1} f(x)\) exists and is equal to \(1\). This step is crucial when checking continuity at a point for a piecewise function.

3. Comparing function value and limit: We have \(f(1) = 1\) and \(\lim_{x \to 1} f(x) = 1\).

   Since \(\lim_{x \to 1} f(x) = f(1)\), the function \(f(x)\) is continuous at \(x = 1\). This means it behaves like a continuous function around \(x=1\).

Conclusion on Function Continuity

Based on our analysis, the function \(f(x)\) is continuous at \(x = 0\) and also continuous at \(x = 1\). Therefore, the function \(f(x)\) is a continuous function at both transition points.

This detailed step-by-step analysis helps in understanding how to determine the continuity at a point for a piecewise function. Calculating the limit and checking the function value are key steps for checking continuity at a point.

In summary, the given function is continuous at both \(x=0\) and \(x=1\).