An infinitely differentiable function that is not analytic (3215 bytes)
9: ...f''^(''n'') we prove to exist. For any differentiable function ''R''(''x'') the derivative for ''x'' > ...
34: ...rentiable function one can get another infinitely differentiable function with prescribed behavior <!-- prescribed...
36: ...that the sheaf of differentiable functions on a [[differentiable manifold]] is [[flasque]], in contrast with the a...
38: ... up [[partition of unity|partitions of unity]] on differentiable manifolds.

Green's identities (1435 bytes)
5: ... differentiable]], and &psi; is once continuously differentiable, on some region ''U'', then:
10: If &phi; and &psi; are both twice continuously differentiable on ''U'', then:
15: If &psi; is twice continuously differentiable on ''U''

Weierstrass function (704 bytes)
1: .... [[Almost all]] continuous functions are nowhere differentiable, and this property is both stable and generic. We...

Piecewise (1621 bytes)
5: ...d. A [[path]] said to be ''piecewise continuously differentiable'' is a ''continuous'' path (in the plane, say) bu...

Critical value (660 bytes)
5: ...al value''' of a [[differentiable map]] between [[differentiable manifold]]s is the image of a [[critical point]].

The analytization trick (1839 bytes)
3: ...analytic function|analytic]], but happens to be [[differentiable]] with respect to its [[real number|real]] and [[...
20: ...r|real]] and [[imaginary]] components (or n times differentiable, as the case may be). In that case,

Directional derivative (1872 bytes)
1: ... the [[directional derivative]] of a multivariate differentiable function along a given unit vector intuitively re...
4: The [[directional derivative]] of a differentiable function <math>f(\vec{x}) = f(x_1, x_2, \ldots, x...
15: ...th> by evaluating the directional derivative of a differentiable function <math>f</math> along the unit vector <ma...

Holomorphic function (4472 bytes)
1: ...'a''" means not just differentiable at ''a'', but differentiable everywhere within some open disk centered at ''a'...
5: ...''' is a function, we say that ''f'' is ''complex differentiable'' at the point ''z''₀ of ''U'' if the ...
12: Intuitively, if ''f'' is complex differentiable at ''z''₀ and we approach the point ''...
15: If ''f'' is complex differentiable at ''every'' point ''z''₀ in ''U'', we...
32: ... holomorphic function is infinitely often complex differentiable at every point. It coincides with its own [[Taylo...

Volterra's function (2639 bytes)
3: * <math>V(x)</math> is [[differentiable]] everywhere
11: ...th-Volterra-Cantor set|SVC]], but the function is differentiable at these points with value 0. Furthermore, in an...

Concavity (3059 bytes)
5: ...ex function has an increasing slope. Similarly, a differentiable [[graph of a function|function]] ''f'' is '''conc...
9: For a twice-differentiable function ''f'', if the second derivative, ''f''<n...
30: If ''f''(''x'') is twice-[[differentiable]], then ''f''(''x'') is concave [[iff]] ''f'' &pr...

Lie derivative (2488 bytes)
9: ...f [[rank of a tensor|rank]] (''p'' ''q'') (i.e. a differentiable [[linear map]] of [[smooth function|smooth]] [[se...
25: and a differentiable [[vector field]] (section of the [[tangent bundle...

Real line (2693 bytes)
13: ...ard differentiable structure on it, making it a [[differentiable manifold]].
14: (Up to [[diffeomorphism]], there is only one differentiable structure that the topological space supports.)

Rolle's theorem (3291 bytes)
2: ... a closed interval [''a'',''b''] and [[derivative|differentiable]] on the open interval (''a'',''b''), and ''f''('...
19: ... non-negative. (Note that we assume that ''f'' is differentiable to guarantee that the left-hand and right-hand de...
23: Finally, since ''f'' is differentiable at ''x'', these two limits must be equal and henc...

Diffeology (3783 bytes)
12: ...e]], and its [[inverse function|inverse]] is also differentiable.
14: The diffeological spaces, together with differentiable maps as morphisms, form a [[category theory|categ...

Affine connection (346 bytes)
1: ...les|connection]] on the [[tangent bundle]] of a [[differentiable manifold]]. It can either be torsionless or have ...

Lipschitz continuity (2100 bytes)
24: ... Conversely, if ''f'' : ''I'' &rarr; '''R''' is a differentiable map with bounded derivative, |<em>f'</em>(''x'')|...

Smooth (296 bytes)
3: ...[Smooth function]], a function that is infinitely differentiable, used in [[calculus]] and [[topology]].

Lie groupoid (291 bytes)
1: ...d target functions]] are nondegenerate [[Manifold|differentiable functions]]. It is a generalization of a [[Lie gr...

Product rule (6916 bytes)
3: ...the [[derivative|differentiation]] of products of differentiable [[function (mathematics)|function]]s.
15: ...rgument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv...
34: ...fferentiable function, then ''cf''(''x'') is also differentiable, and its derivative is (''c'' &times; ''f'')<nowi...
48: and suppose further that ''g'' and ''h'' are each differentiable at the fixed number ''x''. Then
99: ...continuous]] [[bilinear operator]]. Then ''B'' is differentiable, and its derivative at the point (''x'',''y'') in...

Proof that holomorphic functions are analytic (3350 bytes)
4: ...tion|holomorphic]] at a point ''a'' [[iff]] it is differentiable at every point within some open disk centered at ...
16: * the fact that holomorphic functions are differentiable not just once, but infinitely often, and
22: Suppose ''f'' is differentiable everywhere within some open disk centered at ''a'...

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