| Function | At x=0 | Continuous? | Differentiable? | f'(0) |
|---|---|---|---|---|
| |x| | Corner | Yes | No | DNE |
| x|x| | Smooth | Yes | Yes | 0 |
| x^2|x| | Smooth | Yes | Yes (twice) | 0 |
| x sin(1/x) | Oscillating | Yes | No | DNE |
| x^2 sin(1/x) | Oscillating | Yes | Yes | 0 |
| x^3 sin(1/x) | Oscillating | Yes | Yes | 0 |
| [x] | Jump | No | No | DNE |
| {x} | Jump | No | No | DNE |
| sgn(x) | Jump | No | No | DNE |
| x^(1/3) | Vertical tangent | Yes | No | infinity |
| x^(2/3) | Cusp | Yes | No | +/-inf |
| e^(-1/x^2), f(0)=0 | Flat | Yes | Yes | 0 |
Part of CALC-08 — Continuity & Differentiability (Advanced)
Special Functions Differentiability Chart
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