Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A differentiable function is a function whose derivative exists at each point in its domain. This means the function is both continuous and smooth, with no sharp corners or cusps.
5 Must Know Facts For Your Next Test
A function is differentiable at a point if it has a finite derivative at that point.
If a function is differentiable at every point in an interval, it is said to be differentiable on that interval.
Differentiability implies continuity: if $f$ is differentiable at $a$, then $f$ is continuous at $a$.
The converse is not true: a function can be continuous but not differentiable (e.g., $|x|$).
The derivative of a differentiable function gives the slope of the tangent line to the curve at any given point.
Review Questions
Related terms
Continuous Function: A function that does not have any abrupt changes in value; it can be drawn without lifting your pen from the paper.
Derivative: A measure of how a function changes as its input changes; formally, it represents the slope of the tangent line to the graph of the function.
$\lim$ (Limit): $\lim_{x \to c} f(x)$ describes the behavior of $f(x)$ as $x$ approaches some value $c$. Limits are foundational for defining derivatives and continuity.