Definition D1
That f is differentiable at x means the limit lim (h→0) [f(x+h) − f(x)]/h exists, and its value is written f′(x).
Proof.
(1)a − b means a + (−b), and F/h means F · h⁻¹. The difference quotient is defined only for h ≠ 0.convention
(2)If the limit exists, its value is unique. Hence the notation f′(x) is justified.A6
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Definition D2
That f is continuous at x means lim (h→0) f(x+h) = f(x).
Proof.
This means two things at once: the limit on the left exists, and its value equals f(x).convention
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Theorem T1
If f is differentiable at x, then f is continuous at x.
Proof.
(1)Suppose f is differentiable at x.assumption, D1
(2)If h ≠ 0, then f(x+h) − f(x) = h · {[f(x+h) − f(x)] · h⁻¹}A8, A9, A10, A11
(3)lim (h→0) [f(x+h) − f(x)]/h = f′(x)(1), D1
(4)lim (h→0) h = 0A3
(5)Taking limits of both sides of (2) gives lim (h→0) [f(x+h) − f(x)] = 0 × f′(x) = 0(2), (3), (4), E, A2, A12
(6)f(x) + [f(x+h) − f(x)] = f(x+h)A14, A15, A16, A17
(7)lim (h→0) f(x+h) = f(x) + 0 = f(x)(5), (6), A1, A4, A17
(8)Therefore f is continuous at x.(7), D2
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Corollary C
The converse does not hold. g(x) = |x| is continuous at 0 but not differentiable at 0.
Proof.
(1)|t| means t if t ≥ 0 and −t if t < 0. Hence |0| = 0.convention
(2)If h > 0, then |0 + h| − |0| = h; if h < 0, then |0 + h| − |0| = −h.(1), A15, A16, A17
(3)The right limit is lim h = 0, and the left limit is lim(−h) = (−1) × 0 = 0.(2), A3, A5, A13, A12
(4)The two values agree, so lim (h→0) |0 + h| = |0|, and g is continuous at 0.(3), A7, D2
(5)If h > 0, then (|0+h| − |0|)/h = h · h⁻¹ = 1.(2), A10
(6)If h < 0, then (|0+h| − |0|)/h = (−1)(h · h⁻¹) = −1.(2), A13, A8, A10
(7)The difference quotient is the constant 1 for h > 0 and the constant −1 for h < 0, so the right limit is 1 and the left limit is −1.(5), (6), A4
(8)1 ≠ −1, so the one-sided limits differ and the limit of the difference quotient does not exist.(7), A18, A7
(9)Therefore g is not differentiable at 0.(8), D1
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