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Why Is A Function Not Differentiable At Corner Point?

Published Aug 29, 2025 6 min read
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A function is not differentiable at a corner point because the derivative, which represents the instantaneous slope of the function, is undefined at that specific location.

The fundamental reason is that the slope of the function as it approaches the corner from the left is different from the slope as it approaches from the right. For a function to be differentiable at a point, the left-hand and right-hand derivatives must be equal, a condition that a corner point inherently violates.

The intuitive understanding: The concept of a unique tangent line

A core geometric interpretation of a derivative is that it represents the slope of the tangent line to the graph at a given point.

  • For a smooth curve, as you "zoom in" on any point, the curve looks more and more like a straight line. A single, unique tangent line can be drawn that just touches the curve at that point.
  • At a sharp corner, this fails. No matter how much you zoom in on the corner, the graph retains its V-shape. It never approximates a single straight line.
  • Because there is no unique tangent line at a corner, the derivativeβ€”the slope of that non-existent tangent lineβ€”cannot be defined. Instead, a corner allows for an infinite number of lines that pass through the point without crossing the curve, none of which can be considered a unique tangent.

The formal explanation: The limit definition of the derivative

The formal definition of the derivative of a function f(x)f of x

𝑓(π‘₯)

at a point x=ax equals a

π‘₯=π‘Ž

is given by the limit of the difference quotient:fβ€²(a)=limhβ†’0f(a+h)βˆ’f(a)hf prime of a equals limit over h right arrow 0 of the fraction with numerator f of open paren a plus h close paren minus f of a and denominator h end-fraction

𝑓′(π‘Ž)=limβ„Žβ†’0𝑓(π‘Ž+β„Ž)βˆ’π‘“(π‘Ž)β„Ž

For this limit to exist, the function must approach the same value regardless of whether hh

β„Ž

approaches 0 from the positive side (from the right) or the negative side (from the left). These are known as the right-hand and left-hand derivatives, respectively.

For the derivative fβ€²(a)f prime of a

𝑓′(π‘Ž)

to exist, the following condition must be met:limhβ†’0βˆ’f(a+h)βˆ’f(a)h=limhβ†’0+f(a+h)βˆ’f(a)hlimit over h right arrow 0 raised to the negative power of the fraction with numerator f of open paren a plus h close paren minus f of a and denominator h end-fraction equals limit over h right arrow 0 raised to the positive power of the fraction with numerator f of open paren a plus h close paren minus f of a and denominator h end-fraction

limβ„Žβ†’0βˆ’π‘“(π‘Ž+β„Ž)βˆ’π‘“(π‘Ž)β„Ž=limβ„Žβ†’0+𝑓(π‘Ž+β„Ž)βˆ’π‘“(π‘Ž)β„Ž

At a corner point, this condition is not satisfied. The left-hand and right-hand limits of the difference quotient are unequal, which means the overall limit does not exist.

Example: The absolute value function

The function f(x)=|x|f of x equals the absolute value of x end-absolute-value

𝑓(π‘₯)=|π‘₯|

is the canonical example of a function with a corner point. Let's examine its differentiability at x=0x equals 0

π‘₯=0

.

  1. **Right-hand derivative at x=0x equals 0

    π‘₯=0** : For h>0h is greater than 0

    β„Ž>0

    , we have |x+h|=|h|=hthe absolute value of x plus h end-absolute-value equals the absolute value of h end-absolute-value equals h

    |π‘₯+β„Ž|=|β„Ž|=β„Ž

    .limhβ†’0+f(0+h)βˆ’f(0)h=limhβ†’0+|0+h|βˆ’|0|h=limhβ†’0+hh=limhβ†’0+1=1limit over h right arrow 0 raised to the positive power of the fraction with numerator f of open paren 0 plus h close paren minus f of 0 and denominator h end-fraction equals limit over h right arrow 0 raised to the positive power of the fraction with numerator the absolute value of 0 plus h end-absolute-value minus the absolute value of 0 end-absolute-value and denominator h end-fraction equals limit over h right arrow 0 raised to the positive power of h over h end-fraction equals limit over h right arrow 0 raised to the positive power of 1 equals 1

    limβ„Žβ†’0+𝑓(0+β„Ž)βˆ’π‘“(0)β„Ž=limβ„Žβ†’0+|0+β„Ž|βˆ’|0|β„Ž=limβ„Žβ†’0+β„Žβ„Ž=limβ„Žβ†’0+1=1

    This shows the slope as we approach the origin from the right is +1positive 1

    +1

    .

  2. **Left-hand derivative at x=0x equals 0

    π‘₯=0** : For h<0h is less than 0

    β„Ž<0

    , we have |x+h|=|h|=βˆ’hthe absolute value of x plus h end-absolute-value equals the absolute value of h end-absolute-value equals negative h

    |π‘₯+β„Ž|=|β„Ž|=βˆ’β„Ž

    .limhβ†’0βˆ’f(0+h)βˆ’f(0)h=limhβ†’0βˆ’|0+h|βˆ’|0|h=limhβ†’0βˆ’βˆ’hh=limhβ†’0βˆ’-1=-1limit over h right arrow 0 raised to the negative power of the fraction with numerator f of open paren 0 plus h close paren minus f of 0 and denominator h end-fraction equals limit over h right arrow 0 raised to the negative power of the fraction with numerator the absolute value of 0 plus h end-absolute-value minus the absolute value of 0 end-absolute-value and denominator h end-fraction equals limit over h right arrow 0 raised to the negative power of negative h over h end-fraction equals limit over h right arrow 0 raised to the negative power of negative 1 equals negative 1

    limβ„Žβ†’0βˆ’π‘“(0+β„Ž)βˆ’π‘“(0)β„Ž=limβ„Žβ†’0βˆ’|0+β„Ž|βˆ’|0|β„Ž=limβ„Žβ†’0βˆ’βˆ’β„Žβ„Ž=limβ„Žβ†’0βˆ’βˆ’1=βˆ’1

    This shows the slope as we approach the origin from the left is -1negative 1

    βˆ’1

    .

Since the left-hand derivative (-1)open paren negative 1 close paren

(βˆ’1)

does not equal the right-hand derivative (1)open paren 1 close paren

(1)

, the derivative of |x|the absolute value of x end-absolute-value

|π‘₯|

at x=0x equals 0

π‘₯=0

does not exist.

Continuity vs. differentiability at a corner

It is important to distinguish between continuity and differentiability.

  • Continuity: A function is continuous at a corner point. The graph can be drawn without lifting your pencil, as there are no breaks or jumps. In the case of f(x)=|x|f of x equals the absolute value of x end-absolute-value

    𝑓(π‘₯)=|π‘₯|

    at x=0x equals 0

    π‘₯=0

    , the limit of the function as xx

    π‘₯

    approaches 0 is 0, and the value of the function at 0 is also 0.

  • Differentiability: Differentiability is a stronger condition than continuity. For a function to be differentiable at a point, it must be continuous, but being continuous is not sufficient. The existence of a corner is a classic case where a function is continuous but not differentiable.

Analogy: The path of a train

Imagine a train traveling along a track.

  • On a smooth, curved section of the track, the train's direction changes gradually. At any single moment, its velocity is defined by a single tangent vector.
  • Now, imagine a track with a very sharp, instantaneous corner. A train could not navigate this corner smoothly. The train's direction would change instantaneously, from moving along one straight track to another. The concept of an "instantaneous velocity" at the precise point of the corner is nonsensical because the direction of travel is undefined at that moment. The derivative of a function at a corner is undefined for the same reason: there is no single instantaneous slope.
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