Differentiable Definition and 290 Threads

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.

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  1. Y

    MHB For which parameter values the function is continuous and differentiable

    For which values of a,b and c, the next function is continuous and differentiable at x=2 ? \left\{\begin{matrix} 3x-1 & x\leq 2\\ ax^{2}+bx+c & x>2 \end{matrix}\right. 1. b=2-c 2. b=6+2c+2a 3. 7+c-2a 4. b=3-a-(3/4)c I know that f(2)=5, and so is the limit of f when x goes to 2 from the left...
  2. J

    Proving that a solution to an IVP is unique and infinitely differentiable

    Homework Statement \frac{d^2y}{dt^2} + t\frac{dy}{dt} + t^3y = e^t;\ \ \ y(0) = 0, \ \ y'(0) = 0 Show that the solution is unique and has derivatives of all orders. Determine the fourth derivative of the solution at t = 0.2. The attempt at a solution I'm somewhat lost here... Trying to...
  3. micromass

    Topology Topology from the Differentiable Viewpoint by Milnor

    Author: John Milnor Title: Topology from the Differentiable Viewpoint Amazon Link: https://www.amazon.com/dp/0691048339/?tag=pfamazon01-20 Prerequisities: Level: Undergrad Table of Contents: Preface Smooth manifolds and smooth maps Tangent spaces and derivatives Regular values The...
  4. C

    Is This Line Integral Differentiable?

    Homework Statement \vec { F } \left( x,y \right) =u\left( x,y \right) \hat { i } +v\left( x,y \right) \hat { j } u\left( x,y \right) , v\left( x,y \right) are continuous on ℝ² \Gamma is piecewise smooth. Is \psi (x,y){ =\int { \vec { F } \left( x,y \right) \cdot \vec { dr } } }...
  5. K

    Differentiable Greatest Integer Function

    Homework Statement k(x)=x2*[1/x] for 0<x≤1 k(x)=0 for x=0 Find where k(x) is differentiable and find the derivative Homework Equations The Attempt at a Solution I know that it is differentiable for all ℝ\Z on (0,1], but I am unsure how to find the derivative for this problem.
  6. C

    How to Determine Complex Differentiability and Holomorphicity of a Function?

    Homework Statement f(z)=z(bar(z))^2+2(bar(z))z^2 ,then calculate the total differential of f viewed as a map from R^2->R^2 . determine the points at which f is complex differentiable , is f holomorhpic anywhere? 2. The attempt at a solution i did the first part and for secund part i use...
  7. Darth Frodo

    F(x) is differentiable at x =1, find f(x)

    Homework Statement suppose a function f is differentiable at x = 1 and lim[h → 0] \frac{f(1 + h)}{h} = 5 Finf, f(1) AND f'(1) The Attempt at a Solution f(x) is differentiable at x = 1 f(x) is continious at x = 1 lim[x→1] f(x)= f(1) f'(x) = lim[h→0] \frac{f(1 + h) -...
  8. Petek

    Continuity of the derivative of a decreasing differentiable function

    Homework Statement To solve a problem in a book, I need to know whether or not the following is true: Let f be a real-valued, decreasing differentiable function defined on the interval [1, \infty) such that \lim_{x \rightarrow \infty} f(x) = 0. Then the derivative of f is continuous...
  9. F

    CR Equations: Real & Imaginary Parts Satisfy Cont. & Diff.

    Complex differentiable <--> real and imaginary parts satisfy C-R eqns and are cont. Say we have a complex function f(z) we can break this into real and imaginary parts: f(z)=u(x,y)+iv(x,y)In my book I am told the following:(1) f complex differentiable at z0 in ℂ --> the Cauchy Reimann...
  10. D

    MHB Continuous periodic piecewise differentiable

    Suppose that $f(\theta)$ is a continuous periodic piecewise differentiable function. Prove that $f(\theta) = f(0) + \int_0^{\theta}g(t)dt$ for a piecewise continuous $g$. I just need a nudge in the right direction here.
  11. P

    Is the function defined, continuous and differentiable

    Homework Statement Graph the function defined by the following. B = {(r/r0)B0 for r ≤ r0 {(r0/r)B0 for r > r0 (a) Is B continuous at r = r0? yes no (b) Is B differentiable at r = r0? Homework Equations The Attempt at a Solution I'm not exactly sure what to do...
  12. T

    Use Taylor's theorem to show a function is differentiable at x=0

    Homework Statement Use Taylor's theorem to estimate |(ex)-x-1| for 0≤x≤1. Thus prove that if a>(1/2) then: f(x)=(1-|x|a)*(ex)a is differentiable at x=0 Homework Equations The Attempt at a Solution So |(ex)-x-1|=(x^2)/2+(x^3)/6+(x^4)/24... But I don't see how this helps, I...
  13. A

    Is the function f(z) = |z|2 differentiable at z0 = a + bi?

    f'(z0)=\stackrel{lim}{x\rightarrow0} \frac{f(z0-z)-f(z0)}{z} Hi, I'm attempting to use the above equation to show where z0 is not differentiable at some point z0 for the equation f(z) = |z|2 I was wondering how I could go about doing this? I tried letting z0 = a + bi, and z = x + yi...
  14. A

    Infinitely differentiable functions

    Suppose I have an infinitely differentiable function F that is nonzero exactly on a set [-b,b]. Can I say that |F(x)| \leq C(x+b)^k for some integer k > 2? If so, why?
  15. I

    Infinitely differentiable function

    This might sound like a stupid question. f(x) = \begin{cases} &e^{-\frac{1}{x^2}} &\text{if } x\neq 0 \\ & 0 &\text{if } x = 0 \end{cases} Is the reason f is infinitely differentiable at 0 because we keep differentiating 0 as a constant, or because, \lim_{x\rightarrow 0} f`(x) =...
  16. N

    Even Differentiable Functions and Linearization

    Is there anything special about even differentiable function of x? Give reasons behind your answer. and Find the linearization of g(x)= 3+ ∫sec(t-1)dt at x=-1 It is a definite intergral going from 1 to x^2.. a=1 b=x^2 I understand how to do regular linearization problems but with this...
  17. C

    Prove that a fuction is continous and differentiable everywhere, but not at f'=0

    Homework Statement Prove that the function f:ℝ→ℝ, given by f(x)={x2sin(1/x) if x≠0, 0 if x=0} is continuous and differentiable everywhere, but that f' is not continuous at 0. Homework Equations The Attempt at a Solution I thought if a function was...
  18. M

    Suppose that F: Rn -> Rn is continuously differentiable everywhere

    Please help me with this. I don't know even how to start Definition: Suppose that F: Rn Rn is continuously differentiable everywhere. A point P∈R^n is called an isolated singularity of F if DF_p is not invertible but DF_y is invertible for all Y≠P in some neighborhood of P. a. Let f: R...
  19. O

    Where is this function differentiable?

    How do I go about determining where f(x,y) = \sqrt{|x| + |y|} is differentiable?
  20. R

    Continously differentiable f: R^n -> R^m not 1-1?

    Continously differentiable f: R^n --> R^m not 1-1? My course is over with now, but I never could figure out this question. It's pretty much been haunting me ever since, and the internet has not given me a proof that convinces me. My problem is determining why: A continuously di...
  21. H

    Proof that a function composition is differentiable

    Homework Statement Let f:R->R, differentiable, f(1)=1 and f'(1)=2. Homework Equations Prove that g:R->R such that g(x)=f(x)Arctg(f(x)) is differentiable in x=1 and calculate g'(1)The Attempt at a Solution I would prove it saying that if a function is differentiable then the product and...
  22. K

    Show that g(x):=|f(x)| is differentiable at c

    Homework Statement Suppose that f:ℝ→ℝ is differentiable at c and that f(c) =0. Show that g(x):=|f(x)| is differentiable at c iff f'(c)=0. The attempt at a solution **This is the solution that I was shown by a peer but I do not understand it... Can anyone break it down for me. It...
  23. R

    Show where the functions is anlaytic and differentiable

    Homework Statement z→x3+ i(1 - y)3: Show where the functions is analytic and differentiable. Homework Equations The Attempt at a Solution For a function to be analytic cauchy-riemann equations must hold.. so ux = vy and uy = -vx Now f(z) = x3 + i(1 - y)3 is already in the...
  24. S

    Is x^0 Differentiable at x=0 Despite 0^0 Being Undefined?

    If the derivative of x^n equals nx^(n-1), then the derivative of x or x^1 equals x^0, but 0^0 is undefined. Does that mean x is not differentiable at zero?
  25. S

    Evaluate where F(x) is differentiable

    Hi there, I cannot seem to figure this question out. Homework Statement Let f: [0,3] -> R be defined as follows x if 0≤x<1, f(X)= 1≤x<2 x if 2≤x≤3 obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where...
  26. O

    MHB Why isn't the "l1"-norm differentiable?

    Hello everyone! I've searched a lot for this one, but couldn't find an answer: If x is in R^N then ||x||_0 = x_1 + x_2 + ... + x_N. Why, then, isn't this norm differentiable? (Btw, how to make LaTeX work?) Thank you!
  27. D

    Linear Algebra. Proving differentiable functions are a vector space.

    Question: Show the set of all differentiable functions on (-infinity, +infinity) that satisfy f′ + 2f = 0 is a vector space. I started the problem by assuming that f and g are both differentiable functions that satisfy this vector space. Then I ran through the ten axioms of addition and...
  28. C

    Is \(\nabla \times (\phi \nabla \phi) = 0\) for a Differentiable Scalar Field?

    How to prove that \nabla x (\phi\nabla\phi) = 0? (\phi is a differentiable scalar field) I'm a bit confused by this "differentiable scalar field" thing...
  29. F

    Limit of Derivatives and Existence of Limits

    Homework Statement can someone help me to see whether the following statements choose or false a) if lim(X->infinite)f(x) exists and is finite and lim(X->infinite)f'(x)=b then b=0 i think it is right but i don't know how to prove it b)if lim(X->infinite)f(x) exists and is finite then...
  30. F

    Function is differentiable at x=0?

    Homework Statement define a function f:R->R is given by f(x)=(|x|^a)*(sin(1/x)) if x is not equal 0 , and f(x)=0 if x=o . and a>0. for which values of a is f differentiable at x=o 2. The attempt at a solution obviously, a=2 is one of the solution , but how about other values? can...
  31. F

    Continuous and differentiable function

    function f:R->R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。then if f is continuous then f1 and f2 may be chosen continuous, and if f is differentiable then f1 and f2 can be chosen differentiable i am quiet confusing this statement , if f1 is continuous f2 is not how their...
  32. F

    Continuous and differentiable function

    Homework Statement function f:R->R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。show that if f is continuous then f1 and f2 may be chosen continuous, and if f is differentiable then f1 and f2 can be chosen differentiable 2. The attempt at a solution i have try some...
  33. J

    Suppose that f is a differentiable real function in

    Homework Statement Suppose that f is a differentiable real function in an open set E (which is a subset of) ℝn, and that that f has a local maximum at a point x in E. Prove that f'(x)=0. Homework Equations Definition. Suppose E is an open set in ℝn, f maps E into ℝm, and x is an...
  34. E

    Partial Derivatives, and Differentiable

    Homework Statement I want to show that the partials exist for a certain function. Homework Equations My book says that if a function f is differentiable at a point x then the partial derivatives exist. The Attempt at a Solution Rather than showing f is differentiable, I am...
  35. A

    Differentiable Automorphisms of ℂ

    Are there any nontrivial differentiable automorphism of the complex numbers? I know there are many automorphisms, but I could only find one article that discussed them. I didn't read the entire thing, but it mentioned that AC is often necessary to construct them, but I didn't see whether it said...
  36. J

    Is showing that a series of functions is differentiable as simple as it appears?

    Here is the problem and my want. I think I might be overlooking something because it seems rather simple...
  37. P

    Differentiable on interval implies monotonic on some neighborhood of every point

    If f is differentiable on [a,b] and f'(c)>0 for some a<c<b then does this imply that f is monotonically increasing on some neighborhood of c? My intuition says yes but I just can't figure out a way to prove it. (not homework). Because of the weierstrass function I'm pretty sure...
  38. P

    Check if the following functions are differentiable

    Homework Statement Check if the following functions f : ℝ → ℝ are differentiable: \displaystyle f(x)=|(x-1)^{2}(x+1)^{3}| \displaystyle f(x)=|x^{2}-\pi^{2}|sin^{2}x Homework Equations The Attempt at a Solution I don't know what the condition should be, I've searched a lot of...
  39. J

    Prove that f is uniformly differentiable

    Homework Statement Suppose f ' is continuous on [a, b] and ε > 0. Prove that there exists ∂ > 0 such that | [f(t)-f(x)]/[t-x] - f '(x) | < ε whenever 0 < |t - x| < ∂, a ≤ x ≤ b, a ≤ t ≤ b. Homework Equations Definitions of continuity and differentiability The Attempt at a SolutionFix x in...
  40. 5

    Prove that Any Cubed Function is Differentiable (delta-epsilon method)

    Homework Statement Prove that: Any function f such that f(x)=x^3 for any x \in R is differentiable. Homework Equations Skip. The Attempt at a Solution Okay! So, to conclude, it must be shown that, for any a in the domain of f , \displaystyle \exists \lim_{h...
  41. G

    Is the following function differentiable?

    I have: http://img12.imageshack.us/img12/6121/capturerhf.png Is the function differentiable in (0,2)? If so, find its Tangent Plane. So far I have We have (\nabla f)(0,2)=(f_x(0,2).f_y(0,2))=\ldots=(0,1) , so if f is differentiable at (0,2) the only possible differential is \lambda...
  42. K

    Prove that, if f(x) is differentiable at x =c , then f(x) is continous at x=c

    Homework Statement From the definition of the derivative, prove that, if f(x) is differentiable at x=c, then f(x) is continuous at x=c. Homework Equations f'(c) = lim [f(x)-f(c)]/(x-c) This is the definition for a function to be differentiable at x->c...
  43. I

    Is f(z)=x differentiable with respect to z?

    Homework Statement The only thing given is f(z)=x. However, I am under the assumption that z is a complex variable where z=x+iy. I'm also assuming that x is a real variable. In this example, I know that f(z)=x is not differentiable with respect to z because it does not satisfy the...
  44. N

    Given a piecewise, prove that it is continous and differentiable

    1. Homework Statement For f(x)= { sin(x)/x if x≠0 , 1 if x=0. (a) Show that f is continuous and differentiable for all x. (b) Show the derivative f'(x) is continous. 2. Homework Equations 3. The Attempt at a Solution I know that if f is differentiable it is continous, so I need...
  45. N

    If the graph of a differentiable function is symmetric

    Homework Statement If the graph of a differentiable function f is symmertic about the line x=a, what can you say about the symmetry of the graph f'? Homework Equations The Attempt at a Solution
  46. P

    Differentiable Function on an interval

    Let f:[a,b]\rightarrowR be continuous on [a,b] and differentiable in (a,b). Show that if lim f'(x)=A as x goes to a then f'(a) exist and equals A. So I was thinking this has to do either with the mean value theorem or Darboux's Theorem. I have that f(b)-f(a)=f'(c)(b-a) by the mean...
  47. E

    Is the function differentiable everywhere?

    I am hoping someone can help me with the following problem: Define f by: f(x, y) = 0 \ if \ (x, y) = (0,0) \ and \ f(x, y) = \frac{xy^{2}}{(x^{2}+y^{4})^{1/2}} \ otherwise The problem is to determine (and prove) whether the function is differentiable everywhere. First of all, the partials...
  48. M

    Real/complex differentiable function

    Homework Statement Is the following function real and complex-differentiable everywhere? f(z) = Re(z) Homework Equations Cauchy-Riemann equations fy = ifx The Attempt at a Solution Let z = x + iy, z1 = x - iy Re(z) can be defined by Re(z) = (z + z1)/2 A function is...
  49. Y

    Analysis: prove that ln(x) is a smooth function (i.e. infinitely differentiable)

    Homework Statement Prove that f(x) is a smooth function (i.e. infinitely differentiable) Homework Equations ln(x) = \int^{x}_{1} 1/t dt f(x) = ln(x) The Attempt at a Solution I was thinking about using taylor series to prove ln(x) is smooth but I'm strictly told to NOT assume f(x) = ln(x)...
  50. N

    Given a piecewise, prove that it is continous and differentiable

    Homework Statement For f(x)= { sin(x)/x if x≠0 , 1 if x=0. (a) Show that f is continuous and differentiable for all x. (b) Show the derivative f'(x) is continous. Homework Equations The Attempt at a Solution I know that if f is differentiable it is continous, so I need to focus...
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