Continuity and differentiability

What it means for a function to be continuous or differentiable at a point, testing piecewise-defined functions at the join, corners, cusps and vertical tangents, and finding constants that make a join smooth.

The two conditions

Testing a piecewise function

Example 1●●

$f(x) = x^2$ for $x < 1$, and $f(x) = 2x - 1$ for $x \ge 1$. Is $f$ continuous and differentiable at $x = 1$?

Values: $1^2 = 1$ on the left and $2(1) - 1 = 1$ on the right. They match, so $f$ is continuous.

Gradients: $2x = 2$ on the left and $2$ on the right. They match, so $f$ is differentiable, with $f'(1) = 2$.

A smooth join

Orange: y = x² for x < 1. Blue: y = 2x − 1 for x ≥ 1. The line is the tangent to the parabola at x = 1, so the join has no corner.

Try this

What would the join look like if the line were $y = 3x - 2$?

Example 2: making it continuous●●

$f(x) = kx^2$ for $x \le 2$, and $f(x) = 3x - 2$ for $x > 2$. Find $k$ so $f$ is continuous. Is it then differentiable?

At $x = 2$: $4k = 6 - 2$, so $k = 1$.

Gradients: $2kx = 4$ on the left, $3$ on the right. They differ, so there is a corner and $f$ is not differentiable at $x = 2$.

Example 3: making it smooth●●●

$f(x) = x^2$ for $x \le 1$, and $f(x) = ax + b$ for $x > 1$. Find $a$ and $b$ so $f$ is differentiable at $x = 1$.

Gradients match: $a = 2 \times 1 = 2$.

Values match: $a + b = 1$, so $b = -1$.

Flashcards

Continuous at $x = a$

Left value $=$ right value $= f(a)$.

Differentiable at $x = a$

Continuous, and left gradient $=$ right gradient.

$y = |x|$ at $0$

Continuous but not differentiable (a corner).

Practice

$f(x) = kx^2$ for $x \le 2$ and $3x - 2$ for $x > 2$. Find $k$ for continuity.
With that $k$, is $f$ differentiable at $x = 2$?
$f(x) = x^2$ for $x \le 1$ and $ax + b$ for $x > 1$ is differentiable. Find $a$.
For the same $f$, find $b$.
At $x = 0$, the function $y = |x|$ is:
Which statement is always true?
$f(x) = x^2$ for $x < 1$ and $2x - 1$ for $x \ge 1$. Find $f'(1)$.
$y = x^{\frac23}$ at $x = 0$ is:
$f(x) = x + 3$ for $x < 2$ and $f(x) = c$ for $x \ge 2$. Find $c$ so $f$ is continuous.