Does continuity imply differentiability?

No. A function can be continuous everywhere yet differentiable nowhere, as Weierstrass showed; differentiability is strictly stronger, requiring a well-defined limiting slope at each point.

What is the difference between continuity and uniform continuity?

Ordinary continuity allows the required closeness to depend on the point, while uniform continuity demands a single tolerance that works across the whole domain, which holds automatically on closed bounded intervals.

Continuity and Differentiation

Continuity captures the idea of a function with no jumps, and differentiation measures its instantaneous rate of change; together they give the rigorous core of single-variable calculus.

Definition

A function is continuous at a point if values near that point map to values near its image; it is differentiable there if its difference quotients approach a limit, the derivative, giving the best local linear approximation to the function.

Scope

This topic covers the epsilon-delta definition of limits and continuity, uniform continuity, the extreme value and intermediate value theorems on compact and connected sets, the definition and rules of the derivative, the mean value theorem, Taylor's theorem with remainder, and L'Hopital's rule.

Core questions

How is continuity defined precisely, and how does uniform continuity strengthen it?
Why do continuous functions on closed bounded intervals attain their extrema and all intermediate values?
What exactly is the derivative, and how does it relate to continuity?
How does the mean value theorem connect a derivative to the global behavior of a function?

Key theories

Intermediate and extreme value theorems: A continuous function on a closed bounded interval takes every value between any two of its values and attains a maximum and a minimum, results that depend on the connectedness and compactness of the interval.
Mean value theorem: A function continuous on a closed interval and differentiable inside it has a point where the derivative equals the average rate of change over the interval, the bridge from local derivatives to global behavior.
Taylor's theorem: A sufficiently differentiable function is approximated near a point by its Taylor polynomial with an explicit remainder term controlling the error, the foundation of local polynomial approximation.

Clinical relevance

Continuity and differentiation justify the modeling tools of science and engineering: derivatives express rates and gradients in physics, Taylor approximation underlies numerical linearization and error estimates, and the extreme value theorem guarantees that optimization problems on compact sets have solutions.

History

Bolzano and Cauchy introduced rigorous definitions of continuity and the derivative in the early nineteenth century, and Weierstrass perfected the epsilon-delta formulation. Weierstrass's example of a continuous but nowhere-differentiable function dispelled the belief that continuity entails differentiability.

Key figures

Augustin-Louis Cauchy
Karl Weierstrass
Bernard Bolzano

Seminal works

rudin1976
bartle2011

Frequently asked questions

Does continuity imply differentiability?: No. A function can be continuous everywhere yet differentiable nowhere, as Weierstrass showed; differentiability is strictly stronger, requiring a well-defined limiting slope at each point.
What is the difference between continuity and uniform continuity?: Ordinary continuity allows the required closeness to depend on the point, while uniform continuity demands a single tolerance that works across the whole domain, which holds automatically on closed bounded intervals.