Differentiable function

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

1. A function is differentiable at a point if it has a finite derivative at that point.
2. If a function is differentiable at every point in an interval, it is said to be differentiable on that interval.
3. Differentiability implies continuity: if $f$ is differentiable at $a$, then $f$ is continuous at $a$.
4. The converse is not true: a function can be continuous but not differentiable (e.g., $|x|$).
5. The derivative of a differentiable function gives the slope of the tangent line to the curve at any given point.

Review Questions

• What conditions must be met for a function to be considered differentiable?
• Explain why every differentiable function must also be continuous.
• Provide an example of a function that is continuous but not differentiable.

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.