Continuity & smoothness as necessary conditions for differentiability or differentiation at a point
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For a function to be differentiable at some point x = a, the function must be smooth at that point.
A function that is smooth at x = a implies:
1. The function is continuous at x = a
2. The gradients of the tangents on either side of x = a are equal
If the function is smooth at that point, then the function is differentiable at that point. In other words, the gradient of the tangent transitions smoothly from the left of x = a to the right of x = a rather than changing abruptly.
While this is not a rigorous definition of smoothness, it is enough at this point to show if a function is differentiable at any given point.
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