5.12 Continuity, differentiability and first principles

The derivative was introduced as the limit of the gradient of a chord (5.1). That definition leaves two matters open: when the limit actually exists, and how it can be used directly to find derivatives. Both depend on a closer look at limits, and at what it means for a graph to be unbroken (continuous) and smooth (differentiable) at a point.

These ideas explain why the differentiation rules work, and also where they stop working. A function can be continuous at a point without having a derivative there, as at the sharp corner of $y = |x|$, and functions defined in pieces, which model situations such as tax bands and parking charges, can fail to be continuous or differentiable at the points where the pieces join.

Limits and their behaviour

The limit of a function at a point

The statement $\lim_{x \to a} f(x) = L$ means that $f(x)$ can be made as close to $L$ as desired by taking $x$

One-sided limits

Sometimes $f(x)$ approaches different values depending on whether $x$ approaches $a$ from below or from above. A one-sided limit looks at only one of these directions.

For $f(x) = \frac{|x|}{x}$, every negative $x$ gives $\frac{-x}{x} = -1$

Convergence and divergence

A limit converges if it equals a finite number $L$. Otherwise it diverges. A limit can diverge in three different ways: the values can grow without bound (towards $+\infty$ or $-\infty$), the two one-sided limits can disagree, or the values can oscillate without settling down. The same words describe sequences and series (1.8), whose sums converge or diverge in the same sense.

Writing $\lim_{x \to 0}\frac{1}{x^2} = \infty$ is a convenient way of describing how the limit diverges, but $\infty$

Continuity

Continuity at a point

Informally, a function is continuous at $x = a$ if its graph can be drawn through $x = a$ without lifting the pen. The formal version makes this precise using limits.

Polynomials, $\sin x$, $\cos x$ and $e^x$ are continuous for every real $x$. Rational functions, $\ln x$ and $\tan x$

Types of discontinuity

A discontinuity is a point where at least one of the three conditions fails. Discontinuities are usually described by how the graph breaks.

A removable discontinuity is named that way because redefining a single value would remove it: setting $f(2) = 4$ for $f(x) = \frac{x^2 - 4}{x - 2}$

Piecewise functions

A piecewise function uses different rules on different parts of its domain. Each piece is usually a continuous function on its own, so the only places where a discontinuity can appear are the joins. At a join $x = a$, compare three numbers: the left-hand limit (from the left piece), the right-hand limit (from the right piece) and $f(a)$ (from whichever piece includes $x = a$). The function is continuous at $a$ when all three are equal.

Real pricing rules are often piecewise, and their jumps have a practical meaning. A car park that charges for each started hour has a cost function that is constant within each hour and jumps at every whole number of hours.

Finding a constant that makes a function continuous

When a piecewise function contains an unknown constant, continuity at the join gives an equation for it: set the left-hand limit equal to the right-hand limit.

Differentiability

Differentiability at a point

The derivative of $f$ at $x = a$ is the limit of the gradient of the chord from $(a, f(a))$ to a nearby point $(a + h, f(a + h))$

Like any limit, it may or may not exist. The quotient $\frac{f(a + h) - f(a)}{h}$ is called a difference quotient; taking $h > 0$ uses chords to the right of $a$

A function is differentiable on an interval if it is differentiable at every point of it. Polynomials, $\sin x$, $\cos x$ and $e^x$ are differentiable everywhere, and $\ln x$ is differentiable for $x > 0$

Corners: the modulus function

The graph of $y = |x|$ is a V shape, with a sharp corner at the origin. To the left of $0$ the graph is the line $y = -x$, with gradient $-1$; to the right it is $y = x$

Cusps and vertical tangents

A cusp is a sharper kind of corner, where the gradients on the two sides tend to $+\infty$ and $-\infty$. The function $y = x^{\frac{2}{3}}$

Differentiable implies continuous

If $f$ is differentiable at $a$, it must also be continuous there. For $h \neq 0$,

As $h \to 0$, the fraction tends to the finite number $f'(a)$ and $h$ tends to $0$, so the product tends to $f'(a) \times 0 = 0$

Differentiability of piecewise functions

For a piecewise function whose pieces are each differentiable, two conditions at a join $x = a$ make it differentiable there:

1. Continuity: the pieces meet, so their values at $a$ are equal.

2. Matching gradients: the derivatives of the two pieces are equal at $a$, so the join is smooth.

The first condition cannot be skipped. Matching gradients alone are not enough, because a function with a jump is never differentiable.

Differentiation from first principles

The definition of the derivative

Replacing the fixed point $a$ by a general $x$ turns the derivative at a point into the derivative function. Finding a derivative directly from this limit is called differentiation from first principles.

The difference quotient cannot simply be evaluated at $h = 0$, because that gives $\frac{0}{0}$. The method is to simplify it algebraically until the factor $h$ in the denominator cancels, and only then let $h \to 0$

1. Write down $f(x + h)$, replacing every $x$ in $f(x)$ by $(x + h)$.

2. Simplify the numerator $f(x + h) - f(x)$

Deriving the derivatives of x² and x³

For $x^3$, the binomial expansion of $(x + h)^3$ does the same job. Every term except the first contains $h$, and after dividing by $h$ every term except one still contains

The same argument works for any positive integer $n$. By the binomial theorem (1.9),

Subtracting $x^n$ and dividing by $h$ leaves $nx^{n-1}$ plus terms that all contain $h$, so the limit is $nx^{n-1}$

Polynomials and constants

First principles also confirm the rules for constant multiples and sums: in the difference quotient, a constant factor stays outside, a constant term cancels in $f(x + h) - f(x)$, and the terms of a sum can be handled separately.

The derivative of 1/x

For a reciprocal, the two fractions in the numerator are combined over a common denominator before dividing by $h$.

Square roots and derivatives at a point

A square root is handled by rationalising the numerator, the same technique as for the limit $\lim_{x \to 0}\frac{\sqrt{x + 9} - 3}{x}$

The derivative of sin x

The derivative of $\sin x$ can also be found from first principles, using the compound angle identity $\sin(x + h) = \sin x\cos h + \cos x\sin h$ (3.10) and two standard limits for $h$

The first of these is studied in 5.13; here both are taken as known. Then

As $h \to 0$, this tends to $\sin x \times 0 + \cos x \times 1 = \cos x$. The two standard limits are only true when $h$

Higher derivatives

Notation

Differentiating the second derivative (5.7) gives the third derivative, and so on. Any derivative found by differentiating more than once is a higher derivative.

The brackets in $f^{(4)}(x)$ distinguish a derivative from a power. Without them, $f^4(x)$ could be confused with the fourth power $(f(x))^4$

Patterns in higher derivatives

For many functions the higher derivatives follow a pattern, which can be written as a formula for $f^{(n)}(x)$. Working out the first three or four derivatives, without simplifying the numbers too far, usually makes the pattern visible.

Alternating signs are produced by $(-1)^n$, and growing products such as $1, 2, 6, 24$ are factorials. For $\frac{1}{x}$

Proving a pattern by induction

A pattern found from the first few derivatives is only a conjecture. Proof by induction (1.15) turns it into a theorem: show the formula for $n = 1$, then show that if it holds for $n = k$, differentiating once more gives the formula for $n = k + 1$.

Exam-style questions on first principles and differentiability

These topics appear mainly on the non-calculator HL paper, often as a short "prove from first principles" question or as a piecewise function with unknown constants. Induction on higher derivatives is a common longer question, and the ideas of continuity and differentiability underpin the investigations on Paper 3.

A longer question often combines higher derivatives with induction. The early parts are routine differentiation, the conjecture must be written as a formula in $n$, and the proof uses the chain rule in the inductive step.

Self-check

• To do:

  I can find left-hand and right-hand limits and decide whether a limit exists.

• To do:

  I can decide whether a limit converges or diverges, and describe how it diverges.

• To do:

  I can state the three conditions for continuity at a point and classify a discontinuity as removable, jump or infinite.

• To do:

  I can check a piecewise function for continuity at a join, and find a constant that makes it continuous.

• To do:

  I can explain why $|x|$ is continuous but not differentiable at $0$, and recognise corners, cusps and vertical tangents.

• To do:

  I can find constants that make a piecewise function differentiable, using continuity and matching gradients.

• To do:

  I can derive the derivatives of $x^2$