Continuity and Differentiability — Part 1: Continuity, discontinuity and differentiability: visual lesson, ISC Class 12 Mathematics

Lesson

Continuity and Differentiability, Part 1

When can you draw a graph without lifting the pen, and when does it have a smooth slope?

In this lesson you will

Test the continuity of a function at a point with one-sided limits
Find constants that make a piecewise function continuous
Classify a discontinuity as removable, jump or infinite, and remove it when possible
Test the differentiability of a function at a point, including mod x and the greatest integer function
Use the fact that differentiability implies continuity

Part 1 at a glance.

Concept

$f$ is continuous at $a$ when $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$ .

Continuity at $a$: three numbers, one value.
Number	Notation	Must
Left-hand limit	$\lim_{x \to a^-} f(x)$	exist (be finite)
Right-hand limit	$\lim_{x \to a^+} f(x)$	exist and equal the left-hand limit
Value	$f(a)$	be defined and equal both limits

Diagram

With $k = 1$ both pieces meet at $(3, 10)$.

$f(x) = kx^2 + 1$ for $x \le 3$ and $4x - 2$ for $x > 3$ meet when $9k + 1 = 10$ .