A function is differentiable at a point if its derivative exists there, meaning you can zoom in on its graph near that point and it looks like a straight line with a well-defined slope.

Core idea (plain language)

Informally, “differentiable at x=ax=ax=a” means:

  • The function has a well-defined instantaneous rate of change at aaa (its derivative f′(a)f'(a)f′(a)).
  • Near x=ax=ax=a, the graph looks smooth and can be closely approximated by a single straight tangent line.

If this happens at every point in its domain, the function is called differentiable on that domain.

Formal definition (single variable)

A function fff is differentiable at x=ax=ax=a if the following limit exists and is finite:

lim⁡x→af(x)−f(a)x−a\lim_{x\to a}\frac{f(x)-f(a)}{x-a}x→alim​x−af(x)−f(a)​

That limit, when it exists, is the derivative f′(a)f'(a)f′(a).

Key consequences:

  • If fff is differentiable at aaa, then fff is also continuous at aaa.
  • The derivative gives the slope of the tangent line to the graph at (a,f(a))(a,f(a))(a,f(a)).

Geometric picture: how the graph looks

When a function is differentiable at a point:

  • The curve is smooth there: no jumps, corners, or cusps.
  • There is no vertical tangent line (which would correspond to infinite slope).
  • If you zoom in enough around that point, the curve becomes almost indistinguishable from a straight line (this is “local linearity”).

When a function is not differentiable at a point, typically at least one of these happens:

  • A jump or other discontinuity.
  • A sharp corner (like the point x=0x=0x=0 on f(x)=∣x∣f(x)=|x|f(x)=∣x∣).
  • A cusp or vertical tangent.

Differentiable vs. continuous

Here is how the two ideas relate.

[7] [1][9] [7] [9][7] [7] [1][7]
Property Continuous Differentiable
Basic meaning No jumps or holes in the graph at the point. Has a well- defined finite derivative (slope) at the point.
Graph behavior Can be drawn without lifting the pen, but may have sharp corners. Graph is smooth with a single tangent line and no sharp corners.
Logical relation Does not guarantee differentiability. Always implies continuity at that point.

Intuition through a tiny story

Imagine driving along a road that represents the graph of a function:

  • If the road is just continuous , there are no gaps, but you might hit a sudden sharp turn.
  • If the road is differentiable , not only is it unbroken, but every bend is smooth; your steering wheel turns gradually, never with an instant, jerky twist.

Mathematically, differentiability captures this smoothness and gives a precise way to talk about how fast things are changing at each instant.

TL;DR:
A function is differentiable at a point if the limit defining the derivative exists and is finite there, which means the graph is smooth and has a well- defined tangent line at that point, and differentiability always implies continuity but not vice versa.

Information gathered from public forums or data available on the internet and portrayed here.