Cusp — Definition, Graph & Examples

Cusp

A sharp point on a curve. Note: Cusps are points at which functions and relations are not differentiable.

A curve on an x-y graph forming a sharp upward point (cusp) at its peak, with both sides sweeping downward and outward.

Key Formula

y^2 = x^3 \quad \Rightarrow \quad y = \pm\, x^{3/2}

Where:

$x$ = The horizontal coordinate (must be ≥ 0 for real values)
$y$ = The vertical coordinate, taking both positive and negative branches

Worked Example

Problem: Show that the curve y = x^(2/3) has a cusp at the origin by examining its derivative.

Step 1: Write the function and find its derivative using the power rule.

y = x^{2/3} \qquad \Rightarrow \qquad \frac{dy}{dx} = \frac{2}{3}\,x^{-1/3} = \frac{2}{3\,\sqrt[3]{x}}

Step 2: Evaluate the derivative as x approaches 0 from the right (positive side).

\lim_{x \to 0^+} \frac{2}{3\,\sqrt[3]{x}} = +\infty

Step 3: Evaluate the derivative as x approaches 0 from the left (negative side).

\lim_{x \to 0^-} \frac{2}{3\,\sqrt[3]{x}} = -\infty

Step 4: Since the function is continuous at x = 0 (y(0) = 0) but the derivative tends to +∞ from one side and −∞ from the other, the curve has a vertical tangent that reverses direction. This creates a cusp at the origin.

\text{Cusp at } (0,\, 0)

Answer: The curve y = x^(2/3) has a cusp at the origin (0, 0). The function is continuous there but not differentiable, and the one-sided derivatives diverge in opposite directions.

Another Example

This example uses a parametric representation instead of an explicit function, showing how cusps arise when both dx/dt and dy/dt vanish simultaneously.

Problem: Determine whether the parametric curve x(t) = t², y(t) = t³ has a cusp, and find its location.

Step 1: Compute dx/dt and dy/dt.

\frac{dx}{dt} = 2t, \qquad \frac{dy}{dt} = 3t^2

Step 2: Find dy/dx using the chain rule for parametric curves.

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} = \frac{3t}{2}

Step 3: Check what happens at t = 0. Both dx/dt and dy/dt equal zero, so dy/dx is the indeterminate form 0/0. The point on the curve is x(0) = 0, y(0) = 0.

\frac{dx}{dt}\bigg|_{t=0} = 0, \quad \frac{dy}{dt}\bigg|_{t=0} = 0

Step 4: Eliminate the parameter: from x = t², we get t = ±√x. Substituting into y = t³ gives y = ±x^(3/2), or equivalently y² = x³. This is the semicubical parabola, which has a cusp at the origin where the two branches meet at a sharp point.

y^2 = x^3 \qquad \Rightarrow \qquad \text{Cusp at } (0,\,0)

Answer: The parametric curve has a cusp at the origin (0, 0). In Cartesian form, the curve is y² = x³, the classic semicubical parabola.

Frequently Asked Questions

What is the difference between a cusp and a corner on a curve?

At a corner, the curve changes direction abruptly and the left-hand and right-hand derivatives exist but are different finite values. At a cusp, the one-sided derivatives both tend to infinity (or one tends to +∞ and the other to −∞), so the tangent line becomes vertical and reverses. Both corners and cusps are points where the function is not differentiable, but cusps have a sharper, pointed appearance.

Why is a function not differentiable at a cusp?

For a function to be differentiable at a point, the limit of the difference quotient must exist as a single finite value. At a cusp, the slopes from the left and right sides diverge to infinity (often in opposite directions), so no single tangent line can be defined. The function is still continuous at the cusp — it simply lacks a well-defined derivative there.

What is the most common example of a cusp?

The most frequently cited example is the curve y = x^(2/3), which has a cusp at the origin. Another classic example is the semicubical parabola y² = x³. Both appear regularly in calculus textbooks to illustrate points of non-differentiability.

Cusp vs. Corner

	Cusp	Corner
Definition	A sharp point where the tangent direction reverses through infinity	A point where the curve changes direction abruptly at a finite angle
One-sided derivatives	Both tend to ±∞ (diverge)	Both exist as finite but unequal values
Classic example	y = x^(2/3) at x = 0	y = \|x\| at x = 0
Differentiable?	No	No
Continuous?	Yes	Yes

Why It Matters

Cusps appear frequently in calculus when you study differentiability — they are one of the key types of points where a derivative fails to exist, alongside corners, vertical tangents, and discontinuities. Recognizing cusps helps you accurately sketch curves and understand the behavior of functions involving fractional exponents. In more advanced courses, cusps arise in the study of parametric curves, algebraic geometry, and even real-world contexts like the shape of a cardioid or the caustic patterns formed by light reflecting inside a cup.

Common Mistakes

Mistake: Confusing a cusp with a vertical tangent. Students sometimes assume any point where the derivative is undefined must be a cusp.

Correction: A vertical tangent occurs when the slope approaches +∞ (or −∞) from both sides in the same direction — the curve passes smoothly through the point. A cusp requires the slope to diverge in opposite directions (one side +∞, the other −∞), creating a sharp reversal. Check both one-sided limits of the derivative.

Mistake: Assuming a cusp means the function is discontinuous at that point.

Correction: A cusp is a point where the function is continuous but not differentiable. The curve reaches the cusp point without any break or gap; it simply forms a sharp point there.

Related Terms

Differentiable — Functions are not differentiable at cusps
Function — Cusps occur on the graphs of functions
Relation — Cusps also appear in implicit relations like y² = x³
Point — A cusp is a specific type of point on a curve
Continuous — Curves are continuous at cusp points
Derivative — The derivative is undefined at a cusp
Tangent Line — No unique tangent line exists at a cusp