Continuity, Differentiability and the Mean Value Theorems

Continuity

$f$ is continuous at $a$ if

$$\lim_{x\to a}f(x) = f(a),$$

which packs three conditions: $f(a)$ is defined, the limit exists, and they are equal. $f$ is continuous on an interval if it is continuous at every point (one-sided at closed endpoints).

Polynomials, $e^x$, $\sin$, $\cos$ are continuous everywhere; rational functions, $\ln$, $\sqrt{\ }$ and $\tan$ are continuous on their domains. Sums, products, quotients (where the denominator $\ne 0$) and compositions of continuous functions are continuous.

Types of discontinuity

Two theorems about continuous functions on $[a, b]$

• Intermediate value theorem: $f$ takes every value between $f(a)$ and $f(b)$. In particular, if $f(a)$ and $f(b)$ have opposite signs, $f$ has a root in $(a, b)$ — the basis of the bisection method.
• Extreme value theorem: $f$ attains a maximum and a minimum on $[a, b]$. (Fails on open intervals: $1/x$ on $(0, 1)$.)

The derivative

$$f'(a) = \lim_{h\to0}\frac{f(a + h) - f(a)}{h},$$

the slope of the tangent line and the instantaneous rate of change. $f$ is differentiable at $a$ if this limit exists (left and right derivatives exist and agree).

Differentiable ⇒ continuous, never the reverse. Places where a continuous function fails to be differentiable: corners ($\lvert x\rvert$, ReLU at 0), cusps ($x^{2/3}$), vertical tangents ($x^{1/3}$).

Differentiation rules

The chain rule is the mathematical core of backpropagation: the gradient of a composition is the product of local derivatives along the path — see neural networks.

Implicit differentiation: differentiate both sides of $F(x, y) = 0$ treating $y$ as $y(x)$. For $x^2 + y^2 = 25$: $2x + 2yy' = 0 \Rightarrow y' = -x/y$.

Higher derivatives $f''$, $f'''$, … describe curvature and appear in Taylor series and the second-derivative test.

Rolle's theorem and the mean value theorem

Rolle: if $f$ is continuous on $[a, b]$, differentiable on $(a, b)$ and $f(a) = f(b)$, then $f'(c) = 0$ for some $c \in (a, b)$.

Mean value theorem (Lagrange): under the same smoothness conditions,

$$f'(c) = \frac{f(b) - f(a)}{b - a}\quad\text{for some } c\in(a, b).$$

Somewhere the instantaneous slope equals the average slope. Consequences:

• $f' = 0$ on an interval ⇒ $f$ is constant; $f' > 0$ ⇒ strictly increasing; $f' < 0$ ⇒ strictly decreasing.
• $\lvert f(b) - f(a)\rvert \le M\lvert b - a\rvert$ if $\lvert f'\rvert \le M$ (Lipschitz bounds, used in convergence proofs for gradient descent).

Cauchy's MVT: $\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$ — the theorem behind L'Hôpital's rule.

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