Jump Discontinuities (When the Limit Does Not Exist)

What a jump discontinuity is

A jump discontinuity occurs when the function approaches two different values from the left and from the right of a point.

Formal notation

$\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$

The video sketches this: a curve coming in towards $x = 2$ from the left and settling at one height, and a separate branch leaving $x = 2$ from the right at a different height — a visible gap between the two.

Worked example

$f(x) = \begin{cases} x + 1 & x < 2 \\ x^2 & x \geq 2 \end{cases}$

This is a piecewise function, so each side uses a different rule.

The method used throughout: always try direct substitution first — if the piece is a polynomial or otherwise continuous, you get a real number and can stop there.

Left-hand limit. Approaching from $x$ values less than $2$, the function is $x + 1$:

$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x + 1) = 2 + 1 = 3$

Right-hand limit. Approaching from $x$ values greater than $2$, the function is $x^2$:

$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} x^2 = 2^2 = 4$

Conclusion. The one-sided limits disagree:

$\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x) \implies \lim_{x \to 2} f(x) \ \text{DNE}$

and because they disagree by a finite gap, this is a jump discontinuity.