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Continuously Differentiable Functions (C¹)

A function is continuously differentiable (class $C^1$ ) if its derivative $f'(x)$ exists at every point AND the derivative function $f'(x)$ is itself continuous. This is a stronger requirement than merely being differentiable — the derivative must vary smoothly without jumps.

Why does it matter that the derivative be continuous? If $f'$ has a jump discontinuity at some point, then locally the function has "different slopes from the left and right" — a kink. Kinks cause problems in optimization algorithms. $C^1$ functions are "smooth enough" for most gradient-based methods to work reliably.

Notation: $C^1(D)$ denotes the class of functions with continuous first derivative on domain $D$ . $C^k(D)$ means the first $k$ derivatives all exist and are continuous. $C^\infty$ means all derivatives exist and are continuous — these are called "smooth" functions.

Formal View

Definition 10.12 — Class C¹

A function

f: D \to \mathbb{R}

is of class

C^1

(or continuously differentiable) on

D

f'(x)

exists for all

x \in D

and

f'

is continuous on

D

. More generally,

f \in C^k

if all derivatives

f', f'', \ldots, f^{(k)}

exist and are continuous.

$C^\infty$ (smooth) functions include polynomials, $\sin$ , $\cos$ , $e^x$ , $\ln x$ . $|x|$ is $C^0$ but not $C^1$ (derivative has a jump at 0). $x|x|$ is $C^1$ but not $C^2$ .

Why This Matters

$C^1$ is the standard regularity assumption for gradient-based optimization — it ensures the gradient is well-defined and varies predictably.

Gradient descent requires $f \in C^1$ : the gradient must be defined and continuous for standard convergence guarantees.
Finite element methods: solution regularity ( $C^k$ ) determines approximation error rates.
PDE theory: elliptic equations have $C^\infty$ solutions under smoothness assumptions on coefficients.

Learning Resources

Smooth functions and differentiability classes

MIT OpenCourseWare

Overview of $C^k$ function classes and their role in analysis.

17 min

Differentiability and continuity

Khan Academy

The relationship between continuity and differentiability.

8 min

Quiz

Question 1

Which statement correctly describes a $C^1$ function?

Question 2

$f(x) = |x|$ is in class $C^1$ .

Common Mistakes

Thinking differentiable everywhere implies $C^1$ — the derivative must also be CONTINUOUS for $C^1$ .
Confusing $C^k$ (first $k$ derivatives continuous) with having $k$ derivatives that are just bounded.