Show that `f(x) = |x-5|` is continuous but not differentiable at `x = 5`.

Explanation

To show that the function \( f(x) = |x - 5| \) is continuous but not differentiable at \( x = 5 \), we will follow these steps:

### Step 1: Check for Continuity at \( x = 5 \)

To determine if \( f(x) \) is continuous at \( x = 5 \), we need to verify that:
\[ \lim_{x \to 5} f(x) = f(5) \]

#### Step 1.1: Calculate \( f(5) \)
\[ f(5) = |5 - 5| = |0| = 0 \]

#### Step 1.2: Calculate the Left-Hand Limit
The left-hand limit as \( x \) approaches 5 from the left (\( x \to 5^- \)):
\[ \lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} |x - 5| = \lim_{x \to 5^-} -(x - 5) = \lim_{x \to 5^-} (5 - x) = 5 - 5 = 0 \]

#### Step 1.3: Calculate the Right-Hand Limit
The right-hand limit as \( x \) approaches 5 from the right (\( x \to 5^+ \)):
\[ \lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} |x - 5| = \lim_{x \to 5^+} (x - 5) = 5 - 5 = 0 \]

#### Step 1.4: Conclusion for Continuity
Since both the left-hand limit and the right-hand limit equal \( f(5) \):
\[ \lim_{x \to 5} f(x) = 0 = f(5) \]
Thus, \( f(x) \) is continuous at \( x = 5 \).

### Step 2: Check for Differentiability at \( x = 5 \)

To check if \( f(x) \) is differentiable at \( x = 5 \), we need to calculate the left-hand derivative and the right-hand derivative.

#### Step 2.1: Calculate the Left-Hand Derivative
The left-hand derivative at \( x = 5 \) is given by:
\[ f'(5^-) = \lim_{h \to 0} \frac{f(5 - h) - f(5)}{-h} \]
Substituting the values:
\[ f'(5^-) = \lim_{h \to 0} \frac{f(5 - h) - 0}{-h} = \lim_{h \to 0} \frac{-(5 - h - 5)}{-h} = \lim_{h \to 0} \frac{h}{-h} = -1 \]

#### Step 2.2: Calculate the Right-Hand Derivative
The right-hand derivative at \( x = 5 \) is given by:
\[ f'(5^+) = \lim_{h \to 0} \frac{f(5 + h) - f(5)}{h} \]
Substituting the values:
\[ f'(5^+) = \lim_{h \to 0} \frac{f(5 + h) - 0}{h} = \lim_{h \to 0} \frac{(5 + h - 5)}{h} = \lim_{h \to 0} \frac{h}{h} = 1 \]

#### Step 2.3: Conclusion for Differentiability
Since the left-hand derivative and the right-hand derivative are not equal:
\[ f'(5^-) = -1 \quad \text{and} \quad f'(5^+) = 1 \]
Thus, \( f(x) \) is not differentiable at \( x = 5 \).

### Final Conclusion
We have shown that the function \( f(x) = |x - 5| \) is continuous at \( x = 5 \) but not differentiable at that point.

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