Why partial derivatives are not enough to define multivariable smoothness
In short: In single-variable calculus, a differentiable function must be continuous. In the multivariable world, this rule breaks down. This video explores why having partial derivatives doesn't guarantee continuity and what that means for how we define change in higher dimensions.
Dr. Trefor Bazett demonstrates that while differentiability implies continuity in single-variable calculus, the same logic does not apply to multivariable functions. He introduces a specific example called the "cross function" to prove that a function can possess both partial derivatives at a point while remaining discontinuous.
This discrepancy highlights a fundamental gap in multivariable calculus: simply knowing how a function changes along the x and y axes independently is insufficient to describe its overall behavior. Because of this, mathematicians must develop a more robust concept of differentiability that goes beyond just having partial derivatives.
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Sources
- Summary by Intellect · source: Continuity vs Partial Derivatives vs Differentiability | My Favorite Multivariable Function
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