What is Differentiability: Definition and 196 Discussions
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f is also called locally linear at x0 as it is well approximated by a linear function near this point.
##(f(c) - f(a))((f)(b) - f(c)) <0## tells us that there are two cases:
##f(c) >f(a), f(b) ##
##f(c) <f(a), f(b) ##.
I guess we need to define a new function here that let us use the Rolle's theorem..
But it is not clear enough how to do so.
I would ask for a clarification about the following definition of tangent vector from J. Lee - Introduction to Smooth Manifold. It applies to Euclidean space ##R^n## with associated tangent space ##R_a^n## at each point ##a \in R^n##.
$$D_v\left. \right|_a (f)=D_vf(a)=\left. \frac {df(a + tv)}...
Hi, I'm keep studying The Road to Reality book from R. Penrose.
In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...
Hi, I'd like to ask for clarification about the definition of differential of a smooth scalar function ##f: M \rightarrow \mathbb R## between smooth manifolds ##M## and ##\mathbb R##.
As far as I know, the differential of a scalar function ##f## can be understood as:
a linear map ##df()##...
I have previously shown that the function series is differentiable at ##x\neq 0##. The series converges uniformly (thus pointwise) on ##\mathbb R## and the term wise differentiated series is uniformly convergent on any interval ##d\leq |x|##, where ##d>0##. Moreover, the terms are continuously...
What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
Hey! :giggle:
I want to prove the following:
If $x_0$ is an inner point of $D$ ($x_0 \in \text{int } D$), so the differentiability of $f$ at $x_0$ is equivalent to each of the following two conditions.
(i) $\exists \alpha\in \mathbb{C}$ : $\forall \epsilon>0 \ \exists \delta>0\ \forall x\in...
In defining the Wirtinger (aka Cauchy-Riemann) linear operators, often used in signal analysis and in proofs of complex derivatives and the Cauchy-Riemann equations, one assumes differentiability in the real sense. This assumption is usually seen as obvious in the complex analysis setting...
Here is this week's POTW:
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Suppose $f : (a,b) \to \Bbb R$ is a convex function. Show that $f$ is differentiable at all but countably many points and the derivative is nondecreasing.
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Remember to read the...
Dear all,
The function f(x) is defined below:
\[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\]
I want to find for which values of a and b the function is differential at x = 1.
The test I was given, is to check the continuity of both f(x) and f'(x). This is...
Suppose I have an initial condition function ##f(x,t_0 )##, which is everywhere twice differentiable w.r.t. the variable ##x##, but the third or some higher derivative doesn't exist at some point ##x\in\mathbb{R}##.
Then, if I evolve that function with the diffusion equation...
I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ...
The two books are as follows:
"Functions of a Complex Variable...
Many have probably seen an example of a function that is continuous at only one point, for example
##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...
Homework Statement
Examine if the function is differentiable in (0,0)##\in \mathbb{R}^2##? If yes, calculate the differential Df(0,0).
##f(x,y) = x + y## if x > 0 and ##f(x,y) =x+e^{-x^2}*y## if ##x \leq 0 ## (it's one function)
Homework Equations
##lim_{h \rightarrow 0}...
I am reading Reinhold Remmert's book "Theory of Complex Functions" ...I am focused on Chapter 1: Complex-Differential Calculus ... and in particular on Section 2: Complex and Real Differentiability ... ... ...I need help in order to fully understand the relationship between complex and real...
I am reading Reinhold Remmert's book "Theory of Complex Functions" ...
I am focused on Chapter 1: Complex-Differential Calculus ... and in particular on Section 2: Complex and Real Differentiability ... ... ...
I need help in order to fully understand the relationship between complex and real...
In Theodore Shifrin's book: Multivariable Mathematics, he defines the derivative of a multivariable vector-valued function as follows:
Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as...
I have been studying multivariable calculus but I can't quite think visually how a function will be differentiable at a point.
How can a function be differentiable if its partial derivatives are not continuous?
Hello,
I have attached the question and the steps worked out. I am not sure if my steps are correctly. Need advise on that.
Next, I am not sure how to show f''(0) exist or not. Thanks in advance!
Hi,
a basic question related to differential manifold definition.
Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...
Hello,
let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$
be two submanifolds.
We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation
$$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...
Hey! :o
Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be arbitrary and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined by $f(x,y)=yg(x)$.
I want to prove that $f$ is differentiable in the origin if and only if $g$ is continuous in $x=0$.
So that $f$ is differentiable in $(0,0)$ does the...
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 2: Differentiation ... ...
I need help with the proof of Lemma 2.2.3 ... ...
Duistermaat and Kolk's Lemma 2.2.3 and its proof read as follows:
I do not...
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 2: Differentiation ... ...
I need help with understanding an aspect of Definition 2.2.2 ... ...
Duistermaat and Kolk's Definition 2.2.2 reads as...
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 2: Differentiation ... ...
I need help with an aspect of the proof of Proposition 2.2.1 ... ...
Duistermaat and Kolk's Proposition 2.2.1 and its proof...
I want to show that the function defined as follows:
$f(x)=e^{-1/x^2}$ for $|x|>0$ and $f^{(k)}(0)=0$ for $k=0,1,2,\ldots$ is infinitely differentiable but not analytic at the point $x=0$.
For infinite-differentiability I used the fact that $\lim_{|x|\to 0^+} x^{-n} e^{-1/x^2}=0$ for every $n$...
Homework Statement
Let f((x+y)/2)= {[f(x)+f(y)]/2} for all real x and y
{f'(x)=first order derivative of f(x)}
f'(0) exists and is equal to -1 and f(0)=1.
Find f(2)
Homework Equations
Basic formula for differentiablilty:
f'(x)=limit (h tends to 0+) {[f(x+h)-f(x)]/h}
The Attempt at a...
Homework Statement
The function h is differentiable, and for all values of x, h(x)=h(2-x) Which of the following statements must be true?
1. Integral (from 0 to 2) h(x) dx >0
2. h'(1)=0
3.h'(0)=h'(2)=1
A. 1 only
B.2 only
C. 3 only
D. 2 &3 only
E. 1,2 &3
Homework Equations
None that I am...
Sorry, I mistakenly reported my own post last time. But later I realized that these limits do work. So, I'm posting this again.
I'm using these limits to check second-order differentiability:
$$\lim_{h\rightarrow 0}\frac{f(x+2h)-2f(x+h)+f(x)}{h^2}$$
And,
$$\lim_{h\rightarrow...
Hello all,
I wish the find the values of x for which the following function is differentiable:
\[\left | x^{2}-4x+3 \right |\]
I got the point that the function is continuous apart from x=1,3.
I need to find if it is differentiable at x=1,3, using the limit definition of the derivative. I am...
What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...
Homework Statement
Hi everybody! I'm struggling to solve the following problem:
Let ##< \cdot, \cdot >## be an inner product on the vector space ##X##, and ##|| \cdot ||## is the norm generated by the inner product. Prove that the function ##x \in X \mapsto ||x||^2 \in \mathbb{R}## is...
When we talk about differentiability on a
Set X, the set has to be open.
And if a set X is open there exists epsilon> 0 where epsilon is in R.
Then if x is in X, y=x+ or - epsilon and y is also in X
But this contradicts to what i was taught in high school; end points are excluded in the open...
If f and g are continuous functions on the right half-line, [0,∞], then f✶g, the convolution of f and g, is defined by
f✶g(x) = ∫[0,x] f(t)g(x-t)dt.
I would like to know if f✶g is a differentiable function of x.
If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫[0,x]f(t)dt has a derivative...
Homework Statement
I need to see if the function defined as##f(x,y) = \left\{
\begin{array}{lr}
\frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\
0 & (x,y)=(0,0)
\end{array}
\right.##
is differentiable at (0,0)
Homework Equations
[/B]
A function is differentiable at a...
Hello,
Me and my friend were talking about differentiability of some piece-wise functions, but we thought of a problem that we could were not able to come to an agreement on. If the function is:
y=sin(x) for x≠0
and
y=x^2 for x=0,
Is this function differentiable? The graph looks like a normal...
Homework Statement
Assume f:(a,b)→ℝ is differentiable on (a,b) and that |f'(x)| < 1 for all x in (a,b). Let an
be a sequence in (a,b) so that an→a. Show that the limit as n goes to infinity of f(an) exists.
Homework Equations
We've learned about the mean value theorem, and all of that fun...
Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
Alright, so now that I think have some more "mathematical maturity", I have decided to go back and review/re-learn multivariable calculus. I've just started, and have gotten to differentiation.
From what I have seen, most books state the following sufficient condition for differentiability:
A...
Homework Statement
Mod note: Edited the function definition below to reflect the OP's intent.
Suppose f:R->R is continuous. Let λ be a positive real number, and assume that for every x in R and a>0,f(ax)=aλ f(x). (a) If λ > 1 show that f is differentiable at 0. (b) If 0 < λ < 1 show that f is...
This is picture taken from my textbook.
I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous...
Homework Statement
##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##:
The...
Homework Statement
Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?
Homework EquationsThe Attempt at a Solution
f(x) = 1 - x2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0
f' = 2/3 x -1/3.
I don't...
The derivative of ##|f(x)|## with respect to ##x## is ##f'(x)## for ##f(x) > 0## and ##-f'(x)## for ##f(x) < 0##. However, it is undefined wherever the value of the function is zero. I was wondering, though, if the product of this "undefined derivative" and zero is zero.
Homework Statement
I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)}
The problem had three parts.
The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2)
The second was to prove that the function f(x) was continuous...
Homework Statement
Given f(x,y) = x\cdot 3^{x+y^2} . Prove that f is differentiable twice at the point P(1,0).
Homework Equations
D\subset\mathbb{R}^2, f\colon D\to\mathbb{R}, P\in \mathring{D}(interior point) - then f is differentiable n+1 times at P\Leftrightarrow \exists\varepsilon >...
Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2)
Prove that f is differentiable, and find its derivative.
I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?
We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable.
However in the case of 1 independent variable, is it possible for a...
Is there an f(x) which is differentiable n times in a closed interval and (n+1) times in an open interval? I think I saw this in a paper related to Taylor's theorem (could be something else though). It didn't make sense to me, how can something be differentiable more in an interval that contains...
I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function f:M\rightarrow N is independent of the coordinate chart that we use". He shows this is...