Math 2.0

counterexample

Absolute value is continuous but not differentiable at 0

Counterexample

The function f(x)=∣x∣f(x) = |x| is continuous on R\mathbb{R}, but it is not differentiable at 00. The difference quotient is ∣h∣h\frac{|h|}{h}, which is 11 for h>0h > 0 and −1-1 for h<0h < 0. So it has no limit as h→0h \to 0.

Remarks

The one-sided derivatives at 00 exist: the right derivative is 11 and the left derivative is −1-1. So the converse of differentiable functions are continuous fails. For x≠0x \ne 0, the derivative is x/∣x∣x / |x|, the sign of xx. Mathlib states the failure at 00 as not_differentiableAt_abs_zero.

Explanations

L1 human

The graph of ∣x∣|x| is a V with a sharp corner at the origin. At the corner, no single line fits the graph, so there is no tangent.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • edge refutes or not_instance_of out (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 9 nodes in this order. The foundations come first, and the list ends with this node.

Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionOne-sided derivative, definition, no evidence levelOne-sided derivativedefinitionDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementAbsolute value is continuous but not differentiable at 0, counterexample, no evidence levelAbsolute value is con…counterexample
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. Derivative definition
  5. One-sided limit definition
  6. Differentiable function definition
  7. One-sided derivative definition
  8. Differentiable functions are continuous statement
  9. Absolute value is continuous but not differentiable at 0 counterexample

Sources

  1. Web https://en.wikipedia.org/wiki/Absolute_value, Derivative

What links here

No node links to this node yet.