5.12 Continuity, differentiability and first principlesIB Maths: Analysis and Approaches HL: Revision notes

A function is continuous at $x = a$ if its graph has no break there: $f(a)$ is defined and $f(x) \to f(a)$ as $x \to a$ from both sides. Informally, you can draw through $x = a$ without lifting your pen.

A function is differentiable at $x = a$ if the curve has a single, well-defined tangent there, so $f'(a)$ exists.

• $y = x^{2}$ is continuous and differentiable everywhere.
• $y = |x|$ is continuous at $0$ but not differentiable there: the gradient is $-1$ on the left and $+1$ on the right, giving a corner.
• $y = \dfrac{1}{x}$ is not continuous at $0$ (it is not even defined there).

Differentiable $\Rightarrow$ continuous, but continuous does not imply differentiable. At IB you need this informal understanding; you will not be asked to test formally for continuity or differentiability.

$\lim_{x \to a} f(x) = L$ means $f(x)$ gets as close as we like to the single finite value $L$ as $x$ approaches $a$ from both sides. The limit then converges. The value of $f(a)$ itself does not matter — $f$ need not even be defined at $a$: $\lim_{x \to 3}\frac{x^{2} - 9}{x - 3} = \lim_{x \to 3}(x + 3) = 6.$ A limit diverges (does not exist) if $f(x)$ grows without bound, for example $\dfrac{1}{(x-3)^{2}} \to \infty$ as $x \to 3$, or if it approaches different values from the left and right, or if it keeps oscillating.

The same ideas apply as $x \to \infty$: $\dfrac{1}{x} \to 0$ converges, but $x^{2}$ diverges.

The derivative is defined as the limit of the gradient of a chord: $f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}.$ At IB this is used for polynomials only. The method:

1. Write $f(x+h)$ and expand fully.
2. Subtract $f(x)$ — every term without $h$ cancels.
3. Divide every term by $h$.
4. Let $h \to 0$.

Example: $f(x) = 2x^{3} - 5x$. Then $f(x+h) - f(x) = 6x^{2}h + 6xh^{2} + 2h^{3} - 5h$, so the quotient is $6x^{2} + 6xh + 2h^{2} - 5 \to 6x^{2} - 5$.

The difference quotient is an average rate of change over $[x, x+h]$. The derivative is the instantaneous rate of change, the value it converges to as $h \to 0$.

For a ball with height $s(t) = 20t - 5t^{2}$, the average velocity over $[2, 2+h]$ is $-5h$: $-0.5$ for $h = 0.1$ and $0.5$ for $h = -0.1$. Both approach $0$, so the velocity at $t = 2$ is $0$ — the ball is momentarily at rest at the top of its flight.

Differentiating repeatedly gives higher derivatives. The $n$th derivative is written $\dfrac{d^{n}y}{dx^{n}}$ or $f^{(n)}(x)$, so $f'''(x) = f^{(3)}(x)$.

Some functions give a pattern. For $f(x) = x\,e^{2x}$: $f'(x) = (2x+1)e^{2x},\quad f''(x) = (4x+4)e^{2x},\quad f'''(x) = (8x+12)e^{2x},$ suggesting $f^{(n)}(x) = 2^{n-1}(2x + n)e^{2x}$, which can be proved by induction.

For a polynomial of degree $m$, $f^{(m)}$ is a constant and every higher derivative is $0$.