PROOF 134

Prove.

The left and right derivatives must agree

On real functions.

What you may use

A1left limit = right limit ⟺ limit exists
One-sided limits (Proof 39)
A2at most one limit value
Uniqueness of the limit (Proof 39)
A3lim c = c
Limit of a constant function
A4limit exists ⟹ bounded near the point
Local boundedness of convergence (Proof 39)
A5there is exactly one s with s³ = t
Existence of cube roots (completeness)
A60 < a ⟹ 0 < a⁻¹
Reciprocal of a positive number
A70 < c, a < b ⟹ ca < cb
Order and multiplication
A80 < a, 0 < b ⟹ 0 < ab
Product of positives
A9a ≠ 0 ⟹ 0 < a²
Square of a nonzero number
A100 < a ⟹ 0 < a/2 < a
Half of a positive number
A11min(a,b,c) is the smallest of the three
The smallest
A12a < b < c ⟹ a < c
Transitivity of order
A13if a < b then not b ≤ a
Trichotomy of order
A14a ≠ 0 ⟹ a·a⁻¹ = 1
Multiplicative inverse
A15(ab)⁻¹ = a⁻¹b⁻¹
Inverse of a product
A16(ab)c = a(bc)
Associativity of multiplication
A17ab = ba
Commutativity of multiplication
A18a × 1 = a
Multiplicative identity
A19(−1)a = −a
Signs and multiplication
A20a + b = b + a
Commutativity of addition
A21a + (−a) = 0
Additive inverse
A22a + 0 = a
Additive identity
A231 ≠ −1
1 and −1 are distinct numbers
Ea = b ⟹ apply the same operation to both sides
Properties of equality

Nothing outside this list may be cited.

Definition D1

f is differentiable at x means that the limit lim (h→0) [f(x+h) − f(x)]/h exists, and its value is written f′(x).

Proof.

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