Differentiable

For a function to be differentiable, its derivative must exist at every point in its domain, meaning the function is “smooth” and has no sharp corners, breaks, or vertical tangents. Graphically, if you zoom in closely enough on a differentiable function, it will appear as a single, perfectly straight line.

Key requirements for a function to be differentiable include:

No sharp corners: Points that come to a sharp, sudden point (cusp or corner (sharp turn)) are not differentiable because there is no single, clear tangent line.
Must be continuous: A function cannot be differentiable if it has jumps, gaps, or holes (discontinuous). However, being continuous does not automatically mean it is differentiable.
No vertical tangents: A function is not differentiable wherever the slope becomes perfectly vertical (vertical Tangent), as a vertical line has an infinite slope.

Common Examples

Differentiable: f(x) = x² (a smooth, continuous parabola everywhere).
Not Differentiable: f(x) = |x| (the absolute value function). It is perfectly continuous, but it has a sharp, V-shaped corner at x = 0.

Comparison Table

Aspect	Continuous Function	Discontinuous Function
Definition	No interruptions in the graph; satisfies $\lim_{x \to c} f (x) = f (c)$ at every point $c$ .	Has breaks, jumps, or holes; does not satisfy $\lim_{x \to c} f (x) = f (c)$ at some points.
Examples	Polynomials like $f (x) = x^{2} + 3 x - 5$ , trigonometric functions like $f (x) = \sin (x)$ .	Rational functions like $f (x) = \frac{1}{x}$ at $x = 0$ , piecewise functions with mismatched limits.
Limit Behavior	Left and right limits at any point $c$ exist and are equal to $f (c)$ .	At least one point where left and right limits are unequal, do not exist, or do not match $f (c)$ .
Graphical Representation	Unbroken curve; can be drawn without lifting the pencil.	Graph has gaps, jumps, or asymptotes requiring the pencil to be lifted.
Applications	Used in modeling real-world phenomena where changes are smooth and unbroken.	Used to represent scenarios with abrupt changes or undefined behaviors.

Summary and Key Takeaways

Continuity at a point requires the function to be defined, limits to exist, and the limit equals the function value.
Types of discontinuities include removable, jump, and infinite.
- Removable discontinuities are also known as holes. They occur when factors can be algebraically removed or canceled from rational functions.
- Jump discontinuities occur when a function has two ends that don’t meet, even if the hole is filled in at one of the ends.
- Infinite discontinuities occur when a function has a vertical asymptote on one or both sides.
Continuous functions are essential for applying the Intermediate Value Theorem and Extreme Value Theorem.
Understanding continuity is crucial for differentiability and analyzing function behavior.
Graphical interpretation aids in visualizing continuity and identifying discontinuities.

References

“Continuity and Differentiability.” CalcWorkshop, September 6, 2015. https://calcworkshop.com/derivatives/continuity-and-differentiability/.

“Derivatives Tutorial.” NIPISSING UNIVERSITY CALCULUS HELP SITE. Accessed June 30, 2026. https://calculus.nipissingu.ca/tutorials/derivatives.html.

“Differentiable Function.” Wikipedia, June 29, 2026. https://en.wikipedia.org/wiki/Differentiable_function.

“Differentiability of Functions.” GeeksforGeeks, December 2, 2020. https://www.geeksforgeeks.org/maths/differentiability-of-a-function-class-12-maths/.

“Differentiable – Formula, Rules, Examples.” CUEMATH, Accessed June 30, 2026. https://www.cuemath.com/calculus/differentiable/.

Green, Larry. “2.5: Continuity.” LibreTexts, November 2, 2021. https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q1/02%3A_Limits/2.05%3A_Continuity.

Kumaravel, Sujeeth. “Differentiability of a Function Given Its Graph.” Towards AI, August 6, 2023. https://pub.towardsai.net/differentiability-of-a-function-given-its-graph-4f104805036f.

McLean, Rachel. “Differentiable Function: Meaning, Formulas and Examples.” Outlier, March 10, 2022. https://articles.outlier.org/what-does-differentiable-mean.

“Revision Notes – Defining Continuity at a Point.” Sparkl. Accessed June 30, 2026. https://www.sparkl.me/learn/collegeboard-ap/calculus-ab/defining-continuity-at-a-point/revision-notes/885.

“Types of Discontinuity: AP® Calculus AB-BC Review.” Albert Blog & Resources, June 3, 2025. https://www.albert.io/blog/types-of-discontinuity-ap-calculus-ab-bc-review/.

“What does it mean for a function to be differentiable?” SubjectCoach. Accessed June 30, 2026. https://www.subjectcoach.com/tutorials/math/topic/calculus/chapter/what-does-it-mean-for-a-function-to-be-differentiable.

The featured image on this page is from the 11X1 T09 08 implicit differentiation (2010) presentation on the slideshare website.