Continuity and Differentiability Quiz

Continuity in mathematics refers to the absence of any interruption or break in a function's graph. It is significant in the context of differentiability as a function must be continuous at a point in order to be differentiable at that point.

A function must be continuous and smooth (without any sharp corners or cusps) in order to be considered differentiable. Additionally, the function's derivative must exist at every point within its domain.

An example of a function that is continuous but not differentiable is the absolute value function, f(x) = |x|. It is continuous for all real numbers but not differentiable at x = 0 due to the sharp corner at that point.

<p>Derivative = The rate at which a function's value changes with respect to the change in the input variable. Continuity = A function is continuous at a point if the limit of the function at that point exists and is equal to the function's value at that point. Differentiability = A function is differentiable at a point if the derivative at that point exists. Limits = A fundamental concept in calculus that describes the value that a function or sequence approaches as the input or index approaches some value.</p> Signup and view all the answers