Jump Discontinuity | Tony's Intertwingled Memex

Core Definition

A function $f(x)$ has a jump discontinuity at $x = c$ if both the left-hand limit and the right-hand limit exist as finite real numbers, but they are unequal:

\lim_{x \to c^-} f(x) = L_1 \quad \text{and} \quad \lim_{x \to c^+} f(x) = L_2 \quad (\text{where } L_1 \neq L_2)

The jump magnitude is given by

\Delta y = |L_2 - L_1|

. Because the directional limits disagree, the overall two-sided limit

\lim_{x \to c} f(x)

Does Not Exist (DNE).

Concept Explanation

At a jump discontinuity, the graph exhibits a distinct vertical "gap" or "step". Tracing along the curve from the left leads to one height ( $L_1$ ), while tracing from the right leads to a completely different height ( $L_2$ ).

        y ^
          |                 *------------------ (Right path: y -> L2)
       L2 + - - - - - - - - o
          |
       L1 +-----------------* (Left path: y -> L1)
          |                 |
          +-----------------+-----------------> x
                            c

Unlike a removable discontinuity, a jump discontinuity cannot be repaired by redefining a single point, because the incoming paths themselves do not meet.

Algebraic Hallmarks & Engineering Context

Piecewise Function Boundaries: Piecewise formulas that model sudden changes in physical states (such as a switch turning on or a clutch engaging).
Absolute Value Quotients: Functions of the form $\frac{|x - c|}{x - c}$ evaluate to $-1$ for $x < c$ and $+1$ for $x > c$ , generating a jump of magnitude $2$ .
Engineering State Transitions: In electrical and mechanical engineering, jump discontinuities represent instantaneous threshold events:
- Surge protection clamp switching.
- Material tensile fracture (stress dropping instantly from ultimate yield to zero).
- Digital clock pulses and step responses ( $u(t)$ Heaviside functions).

Example Problems

Example Problem 1: Absolute Value Ratio

Evaluate $\lim_{x \to 2^-} f(x)$ , $\lim_{x \to 2^+} f(x)$ , and classify the discontinuity of:

f(x) = \frac{3|x - 2|}{x - 2}

Solution

Step 1: Evaluate left-hand limit ( $x < 2$ )
For $x < 2$ , $|x - 2| = -(x - 2)$ .

\lim_{x \to 2^-} \frac{3[-(x - 2)]}{x - 2} = \lim_{x \to 2^-} (-3) = -3

Step 2: Evaluate right-hand limit ( $x > 2$ )
For $x > 2$ , $|x - 2| = +(x - 2)$ .

\lim_{x \to 2^+} \frac{3(x - 2)}{x - 2} = \lim_{x \to 2^+} (3) = 3

Step 3: Classify
Both directional limits exist and are finite, but $-3 \neq 3$ .
Conclusion: $f(x)$ has a jump discontinuity at $x = 2$ with jump magnitude $\Delta y = |3 - (-3)| = 6$ . The two-sided limit is DNE.

Example Problem 2: Piecewise Boundary Discontinuity

Classify the continuity of $g(x)$ at the boundary $x = 4$ :

g(x) = \begin{cases} 2x + 1 & : x < 4 \\ x^2 - 3 & : x \ge 4 \end{cases}

Solution

Step 1: Compute left-hand limit

\lim_{x \to 4^-} g(x) = \lim_{x \to 4^-} (2x + 1) = 2(4) + 1 = 9

Step 2: Compute right-hand limit

\lim_{x \to 4^+} g(x) = \lim_{x \to 4^+} (x^2 - 3) = 4^2 - 3 = 16 - 3 = 13

Step 3: Compare and conclude
Since $\lim_{x \to 4^-} g(x) = 9 \neq 13 = \lim_{x \to 4^+} g(x)$ , the function has a jump discontinuity at $x = 4$ .

Connections & References

Parent Concepts: Discontinuities, Continuity
Related Concepts: Existence Theorem for Two-Sided Limits, Piecewise Functions, Strategies for One-Sided Limits
Course Links: Calc-1 Session 02 | Calc-1 Session 03