Differentiability Definition and 14 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.

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  1. redtree

    B Difference between a continuously differentiable function and a wave

    What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
  2. hilbert2

    A Function differentiability and diffusion

    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...
  3. hilbert2

    I Idea about single-point differentiability and continuity

    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, &...
    • hilbert2
    • Thread
    • continuity derivative differentiability
    • Replies: 1
    • Forum: Calculus
  4. R

    Differentiability in higher dimensions

    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}...
  5. Jazzyrohan

    I Differentiability of a function of two variables

    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?
    • Jazzyrohan
    • Thread
    • differentiability partial derivatives
    • Replies: 8
    • Forum: Calculus
  6. cianfa72

    B Differentiable function - definition on a manifold

    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...
  7. S

    I Differentiability of multivariable functions

    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...
    • Sho Kano
    • Thread
    • continuity differentiability functions limit multivariable
    • Replies: 9
    • Forum: Calculus
  8. 1

    Differentiability of piece-wise functions

    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...
    • 15adhami
    • Thread
    • calculus 1 differentiability
    • Replies: 4
    • Forum: Calculus
  9. M

    Is ln(x) differentiable at negative x-axis

    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?
  10. Titan97

    Checking if f(x)=g(x)+h(x) is onto

    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...
    • Titan97
    • Thread
    • continuity differentiability functions
    • Replies: 11
    • Forum: General Math
  11. Titan97

    Finding the number of rational values a function can take

    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...
  12. kelvin490

    Differentiability implies continuous derivative?

    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...
  13. W

    Continuity and differentiability in two variables

    Hi If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e. f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2) and f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2) for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##, does it mean that it is continuous...
  14. V

    Allowed values for the "differentiability limit" in complex analysis

    In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit $$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$ But why are the possible ##z_0##'s in the closure of the domain of the original...
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