Mathematics

Continuity only requires the function to have no jumps or breaks—$|x|$ smoothly passes through the origin. Differentiability is stricter: it demands a single, well-defined slope at that point. At $x=0$, the left side of $|x|$ has slope $-1$ while the right side has slope $+1$. Since the slope from the left doesn't match the slope from the right, no single tangent line exists there, so the derivative fails to exist despite the function being continuous.

Continuity only requires the function to have no jumps or breaks—$|x|$ smoothly passes through the origin. Differentiability is stricter: it demands a single, well-defined slope at that point. At $x=0$, the left side of $|x|$ has slope $-1$ while the right side has slope $+1$. Since the slope from the left doesn't match the slope from the right, no single tangent line exists there, so the derivative fails to exist despite the function being continuous.