Question

Show that \( f(x) = |x| \, \textbf{is continuous at} \, x = 0.\)

Hint:

To prove continuity at a point, check that the function is defined at that point and the limit exists and equals the function value.

Answer 1

To show that \( f(x) = |x| \) is continuous at \( x = 0 \), we need to verify that: \[ \lim_{x \to 0} f(x) = f(0) \] 

1. Step 1: Find \( f(0) \): \[ f(0) = |0| = 0 \] 

2. Step 2: Find \( \lim_{x \to 0} f(x) \): We need to evaluate the limit of \( f(x) = |x| \) as \( x \to 0 \). - For \( x > 0 \), \( f(x) = x \). - For \( x < 0 \), \( f(x) = -x \). Thus, the limit of \( f(x) \) as \( x \to 0 \) from both sides is: \[ \lim_{x \to 0^+} f(x) = 0, \lim_{x \to 0^-} f(x) = 0 \] Since both one-sided limits exist and are equal to \( 0 \), we have: \[ \lim_{x \to 0} f(x) = 0 \] 

3. Step 3: Compare the limit with \( f(0) \): Since \( \lim_{x \to 0} f(x) = f(0) = 0 \), the function \( f(x) = |x| \) is continuous at \( x = 0 \). 

Final Answer: Hence, \( f(x) = |x| \) is continuous at \( x = 0 \).