Definition 3.2.1. A function f : S → ℝ is continuous at a point c ∈ S
if for every ε > 0 there exists δ > 0 such that whenever x ∈ S satisfies
|x - c| < δ, then |f x - f c| < ε. A function is continuous on S if it is
continuous at every point of S.
Instances For
A function f : S → ℝ is continuous on S when it is continuous at every
point of S.
Equations
- continuousOnSet S f = ∀ (c : ↑S), continuousAtInSet S f c
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The epsilon-delta continuity at a point agrees with mathlib's notion of continuity for subtype domains.
Continuity on a set agrees with mathlib's Continuous for functions defined
on a subtype of ℝ.
Proposition 3.2.2. For a function f : S → ℝ on a set S ⊆ ℝ and a
point c ∈ S:
(i) if c is not a cluster point of S, then f is continuous at c;
(ii) if c is a cluster point of S, then f is continuous at c if and
only if \lim_{x \to c} f x = f c;
(iii) f is continuous at c if and only if for every sequence
(x_n) ⊆ S with x_n → c, one has f (x_n) → f c.
Example 3.2.3. The reciprocal function f : (0, ∞) → ℝ given by
f x = 1 / x is continuous on its domain.
Proposition 3.2.4. A real polynomial f(x) = a_d x^d + a_{d-1} x^{d-1} + ⋯ + a_1 x + a_0 defines a continuous function ℝ → ℝ.
Proposition 3.2.5. If f, g : S → ℝ are continuous at c ∈ S, then:
(i) the sum x ↦ f x + g x is continuous at c;
(ii) the difference x ↦ f x - g x is continuous at c;
(iii) the product x ↦ f x * g x is continuous at c;
(iv) if additionally g x ≠ 0 for all x ∈ S, the quotient
x ↦ f x / g x is continuous at c.
Example 3.2.6. The sine and cosine functions on ℝ are continuous. The
book justifies this using trigonometric identities and the estimates
|sin x| ≤ |x|, |cos x| ≤ 1, and |sin x| ≤ 1.
Helper lemma for Proposition 3.2.7: composing hf and hg gives the desired
continuity conclusion.
Proposition 3.2.7. If g : A → B is continuous at c ∈ A and
f : B → ℝ is continuous at g c, then the composition f ∘ g : A → ℝ is
continuous at c.
The reciprocal map is continuous on (0, ∞) when regarded as a function
ℝ → ℝ.
Composing the reciprocal with the sine function yields a continuous
function on (0, ∞).
Example 3.2.8. The function x ↦ (sin (1 / x))^2 is continuous on the
open interval (0, ∞).
Proposition 3.2.9. If there exists a sequence (x_n) in S converging to
c such that (f (x_n)) does not converge to f c, then f is
discontinuous at c.
Example 3.2.10. The function f : ℝ → ℝ given by
f x = -1 for x < 0 and f x = 1 for x ≥ 0 has a jump discontinuity at
0. The left-limit along x_n = -1 / n is -1, the right-limit along
x_n = 1 / n is 1, and along the alternating sequence
x_n = (-1)^n / n the values oscillate and diverge.
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Example 3.2.12. The popcorn (Thomae) function on (0, 1) is defined by
f (m / k) = 1 / k when m, k ∈ ℕ are coprime and m / k is in lowest terms,
and f x = 0 when x is irrational. It is continuous at every irrational
c ∈ (0, 1) and discontinuous at every rational c ∈ (0, 1).
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Example 3.2.13. Define g : ℝ → ℝ by g x = 0 for x ≠ 0 and g 0 = 1.
Then g is not continuous at 0 but it is continuous at every other point.
The point 0 is a removable discontinuity since redefining g 0 = 0 would
make the function continuous. Let A = {0} and B = ℝ \ {0}; the restriction
g |_ A is continuous even though g fails to be continuous at 0, while
g |_ B (and g on B) is continuous.