IB Maths AA 5.7 Notes

Continuity & differentiability

In this section, limits are developed further and used to build a deeper understanding of derivatives. The main ideas are continuity and differentiability at a point, limits from first principles, higher derivatives, and evaluating indeterminate limits using l'Hôpital's rule or Maclaurin series.

A function is continuous at a point if there is no break, jump or hole in the graph at that point. Informally, this means that as $x$ approaches a value $a$, the function approaches $f(a)$ itself.

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

If the graph has a hole, jump or vertical asymptote at that point, then the function is not continuous there.

Some examples include:

• $f(x)=x^2$ is continuous for all real values of $x$ because its graph is smooth and unbroken.
• $g(x)=\frac{1}{x}$ is undefined at $x=0$, giving it a hole and making it a discontinuous function.

Additionally, a function is differentiable at a point if it has a well-defined tangent there. Informally, this means the graph is smooth enough to have a unique gradient at that point. If a graph has a sharp corner, cusp or vertical tangent, then it is usually not differentiable there.

The key concept is that a function must be continuous before it can be differentiable, but a continuous function is not always differentiable.

Determine if $f(x)=|x|$ is continuous or differentiable at $x=0$.

It is continuous at $x=0$, but it is not differentiable there because the graph has a sharp corner.