6.3 Limits, Continuity, Differentiability & Mean Value Theorems

Limits: The Working Definition

The CIL paper never asks for an epsilon-delta proof. What it asks is whether a limit exists and what its value is. The operational test is simple: the limit exists at $x = a$ when the left-hand and right-hand limits are finite and equal.

$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \quad \Longrightarrow \quad \lim_{x \to a} f(x) = L$

The standard limits that must be memorised:

Note that $\sin x / x \to 1$ only when $x$ is in radians. A question phrased in degrees is a deliberate trap; the answer becomes $\pi/180$.

Continuity

A function $f$ is continuous at $x = a$ when three conditions hold simultaneously:

1. $f(a)$ is defined,
2. $\lim_{x\to a} f(x)$ exists,
3. $\lim_{x\to a} f(x) = f(a)$.

Discontinuities are classified as removable (the limit exists but disagrees with $f(a)$, so redefining one point repairs it), jump (left and right limits both exist but differ), and infinite (at least one one-sided limit is unbounded).

A practical example from mining instrumentation: a step change in a conveyor load cell signal is a jump discontinuity; a single corrupted sample in an otherwise smooth trace is removable.

Differentiability

The derivative at a point is the two-sided limit

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

and it exists only when the left-hand derivative equals the right-hand derivative. The critical implication chain, tested almost every year in PSU papers, runs one way only:

$\text{differentiable} \Longrightarrow \text{continuous}, \qquad \text{continuous} \nRightarrow \text{differentiable}$

The canonical counter-example is $f(x) = |x|$ at $x = 0$. It is continuous there, but the left-hand derivative is $-1$ and the right-hand derivative is $+1$, so no derivative exists. Geometrically, any sharp corner or cusp destroys differentiability while leaving continuity intact.

Rolle's Theorem

If $f$ is (i) continuous on the closed interval $[a, b]$, (ii) differentiable on the open interval $(a, b)$, and (iii) satisfies $f(a) = f(b)$, then there exists at least one $c \in (a,b)$ with

$f'(c) = 0$

Geometrically: a smooth curve that starts and ends at the same height must be horizontal somewhere in between. All three hypotheses are needed — dropping any one produces a valid counter-example, and CIL-style items often ask precisely which hypothesis fails.

Example. For $f(x) = x^2 - 4x + 3$ on $[1, 3]$: $f(1) = 0$ and $f(3) = 0$, the function is a polynomial so continuity and differentiability are automatic. Then $f'(c) = 2c - 4 = 0$ gives $c = 2$, which indeed lies in $(1,3)$.

Lagrange's Mean Value Theorem

Drop the condition $f(a) = f(b)$ and the conclusion generalises: there exists $c \in (a,b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}$

The instantaneous rate of change somewhere in the interval equals the average rate of change across it. Rolle's theorem is simply the special case in which the right-hand side is zero.

A physical reading familiar from mine haulage: if a dumper covers 30 km in 30 minutes, its average speed is 60 km/h, and the mean value theorem guarantees that its speedometer read exactly 60 km/h at some instant.

Example. For $f(x) = x^2$ on $[1, 4]$, the average slope is $(16 - 1)/(4 - 1) = 5$. Setting $f'(c) = 2c = 5$ gives $c = 2.5 \in (1,4)$.

Cauchy's Mean Value Theorem

The generalisation to two functions states that if $f$ and $g$ satisfy the same hypotheses and $g'(x) \neq 0$ on $(a,b)$, then for some $c$,

$\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$

Setting $g(x) = x$ recovers Lagrange's form. Cauchy's theorem is the formal justification for L'Hopital's rule, which is treated in the next section under indeterminate forms.

Quick Reference: Which Theorem Applies