Why is the existence of two partial derivatives alone insufficient for a function to be differentiable?
Conditions
- The function is evaluated at a reference point .
- The input is displaced by with length .
Reasoning, step by step
- Recall the definition of differentiability: the error must be as .
- Understand that this means .
- Recognize that the existence of partial derivatives only guarantees the slopes in the x and y directions.
- Conclude that without the limit condition on the error, the surface might not have a well-defined tangent plane that approximates it in all directions.
Example
The script states: 'Differentiability means the remaining error divided by the input displacement length tends to zero. Existence of the two partial derivatives alone is insufficient.' The card formula is .
Common misconceptions
- Believing that if partial derivatives exist, the function is automatically differentiable.
- Thinking that the tangent plane is defined solely by the partial derivatives without requiring the error limit condition.
- Assuming that continuity of the function is sufficient for differentiability.
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