Continuity and Differentiability — Part 1: Continuity, differentiability and the chain rule: visual lesson, CBSE Class 12 Mathematics

Lesson

Continuity and Differentiability, Part 1

Can you draw it without lifting the pen? Does it have a slope at every point?

In this lesson you will

Both are unbroken (continuous); only the parabola is smooth (differentiable) at $0$.

Concept

Concept

A function $f$ is continuous at $c$ (a point of its domain) when the limit as $x \to c$ exists and equals $f(c)$ .

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)

All three must hold; a jump fails the middle one.
Condition	Question to ask	At a jump
$f(c)$ defined	Is there a value at $c$ ?	yes
$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$	Do both sides approach the same number?	no
limit $= f(c)$	Does that number equal the value?	—