Question

Prove that the function \( f(x) = |x| \) is continuous at \( x = 0 \) but not differentiable. 

Hint:

To prove continuity, check if the left-hand and right-hand limits exist and are equal. To prove non-differentiability, check if the left-hand and right-hand derivatives match.

Answer 1

Step 1: Continuity of the function at \( x = 0 \) is checked by verifying: \[ \lim_{x \to 0^-} |x| = 0 \quad \text{and} \quad \lim_{x \to 0^+} |x| = 0. \] Since both left-hand and right-hand limits exist and are equal to 0, the function is continuous at \( x = 0 \). 

Step 2: Differentiability is checked by considering the derivative of \( f(x) = |x| \). For \( x>0 \), \( f'(x) = 1 \), and for \( x<0 \), \( f'(x) = -1 \). At \( x = 0 \), the derivative does not exist because the left-hand derivative is not equal to the right-hand derivative. Thus, the function is not differentiable at \( x = 0 \).