The discussion below explains the relationship between limits, continuity and differentiability, together with important examples illustrating when differentiability fails. All mathematical expressions have been formatted using pure HTML so that the article remains fully compatible with Blogger without requiring MathJax, KaTeX or JavaScript.
A derivative is defined using a limit, but the existence of a limit alone does not imply the existence of a derivative.
Notice that this is a very specific limit (called the difference quotient).
A function may have an ordinary limit
while the derivative does not exist.
At x = 0,
exists.
The graph is continuous, but there is a sharp corner at the origin. This geometric feature prevents the derivative from existing at x=0 even though the limit exists.
But
does not exist because
| One-Sided Limit | Value |
|---|---|
| Left-hand limit | −1 |
| Right-hand limit | 1 |
- Left-hand derivative = −1
- Right-hand derivative = 1
- Since the two one-sided derivatives are different, the derivative does not exist.
Hence
| Property | Result |
|---|---|
| Limit exists | ✓ |
| Derivative exists | ✗ |
Your first statement is correct. The existence of an ordinary limit does not guarantee the existence of a derivative. A derivative requires the existence of a very specific limit—the difference quotient—which is a much stronger condition than the ordinary limit.
The proper hierarchy is
Every differentiable function is continuous.
Every continuous function has a limit equal to the function value.
The converse of none of these is true.
Differentiability is the strongest property. Continuity is weaker than differentiability. The existence of a limit is weaker than continuity. Therefore,
(i) Limit exists
↓
(ii) Function may still not be continuous because
may exist but
or f(a) may not even exist.
| { | 1, x = 0 |
| 0, x ≠ 0 |
Here
exists,
but
Hence not continuous.
Every point except the origin lies on the horizontal line y = 0. As x approaches 0 from either side, the function approaches 0. However, the actual value of the function at the origin is f(0)=1, shown by the filled red point. Therefore, the limit exists, but the function is not continuous.
↓
(iii) Even if continuous, the derivative may not exist.
Continuous everywhere.
Not differentiable at 0.
The function is perfectly continuous, yet its graph contains a sharp corner. The left-hand and right-hand slopes are different, so the derivative does not exist at the origin.
Correct.
Differentiability always implies continuity.
Therefore,
If a function is discontinuous, then it cannot be differentiable.
Discontinuous ⇒ Not Differentiable
Correct.
Example
Left slope
Right slope
Hence the derivative does not exist.
Although the graph is continuous, the slope changes abruptly at the corner. Since the left-hand and right-hand derivatives are unequal, the derivative does not exist.
Mostly correct.
Example
The derivative is
At
the derivative becomes infinite.
Most elementary calculus books simply state that the derivative does not exist at this point. Geometrically, however, the tangent line is vertical.
Unlike a corner, the graph remains smooth. However, the tangent becomes perfectly vertical at the origin. Since the slope tends to positive infinity, the ordinary finite derivative does not exist.
Correct.
Example
At 0, define
The function is continuous.
But
Substituting f(h)=h sin(1/h) and f(0)=0,
does not exist because
As h approaches zero, the quantity 1/h becomes arbitrarily large. Consequently, the value of sin(1/h) oscillates endlessly between −1 and 1 without approaching any single value. Therefore, the required limit does not exist.
The function itself approaches the origin continuously. However, its slope oscillates infinitely rapidly near the origin. Since the derivative limit fails to converge, the derivative does not exist.
Let's examine each one.
Correct.
Continuous.
Not differentiable at 0.
Reason: corner.
The function is perfectly continuous because there is no break in its graph. However, the graph changes direction abruptly at the origin. The left-hand derivative is −1 while the right-hand derivative is 1. Since these one-sided derivatives are unequal, the derivative does not exist.
(assuming f(0)=0).
Continuous.
Not differentiable at 0.
Reason: oscillation.
Although the function itself approaches the origin continuously, its slope oscillates increasingly rapidly near the origin. Therefore, the derivative fails to converge even though the function remains continuous.
This is one of the most famous examples.
- Continuous everywhere.
- Differentiable nowhere.
The graph never contains any breaks or jumps, so it is continuous everywhere. However, no matter how much the graph is magnified, it never becomes locally straight. Consequently, no unique tangent exists at any point.
Also called the Takagi function.
- Continuous everywhere.
- Differentiable nowhere.
The Blancmange function exhibits self-similarity at different scales. Although it is continuous everywhere, it has no well-defined derivative at any point.
This one needs a small correction.
The Koch snowflake is not a function. Instead, it is a fractal curve. Therefore, we do not normally discuss its differentiability as a function of the form y = f(x).
More precisely,
- The boundary of the Koch snowflake is a continuous curve.
- It has no well-defined tangent at any point.
- Consequently, it is regarded as a nowhere-differentiable curve.
Unlike the previous examples, the Koch snowflake does not represent a single-valued function. Instead, it is studied as a geometric curve. Its boundary is continuous, but nowhere smooth, making it an important example in fractal geometry.
There is another classic example.
At x = 0, the graph has a cusp.
The derivative approaches
- +∞ from one side,
- −∞ from the other side.
Therefore, the derivative does not exist.
A cusp is fundamentally different from a corner. The graph remains continuous, but the tangent direction changes infinitely rapidly, producing infinite slopes with opposite signs.
Although all three situations result in the derivative not existing, they are geometrically very different. Understanding these differences is essential in elementary calculus.
- Corner: Finite one-sided derivatives exist but are unequal.
- Cusp: One-sided derivatives become infinite with opposite signs.
- Vertical Tangent: Both one-sided derivatives approach the same infinite value.
Although all three graphs are continuous, the derivative fails for different geometric reasons. A corner has two different finite slopes. A cusp has two infinite slopes with opposite signs. A vertical tangent has infinite slopes of the same sign. Recognizing these differences is extremely important when studying differentiability.
The relationship between limits, continuity and differentiability can now be summarized as follows.
Differentiable → Continuous → Limit Exists
Every differentiable function is automatically continuous. Likewise, every continuous function automatically has a limit equal to the function value. However, the converse implications are not true.
A function may possess a limit without being continuous. Similarly, a function may be continuous without being differentiable. Therefore, each property is stronger than the one below it in the hierarchy.
The hierarchy
does not work in the reverse direction.
Many students mistakenly assume that if a function is continuous, it must also be differentiable. The absolute value function, f(x) = |x|, is an immediate counterexample. Likewise, the existence of a limit alone does not guarantee continuity.
A function may fail to be differentiable because of the following reasons.
- Discontinuity ✓
- Corner (sharp turn) ✓
- Cusp ✓
- Vertical tangent ✓
- Wild oscillation ✓
- Fractal behavior (for example, the Weierstrass or Blancmange function) ✓
| Misconception | Reality |
|---|---|
| If the limit exists, the function is continuous. | False. The function value must also equal the limit. |
| If a function is continuous, it is differentiable. | False. The graph may contain a corner, cusp or vertical tangent. |
| Infinite derivative means derivative exists. | False. Elementary calculus requires a finite derivative. |
| Every continuous curve represents a function. | False. The Koch snowflake is a continuous curve but not a function y=f(x). |
- Derivative is defined using a limit.
- Every differentiable function is continuous.
- Every continuous function has a limit equal to its function value.
- The converse of both statements is false.
- Discontinuity always destroys differentiability.
- Corner → unequal finite slopes.
- Cusp → opposite infinite slopes.
- Vertical tangent → same infinite slope.
- Oscillation → derivative limit fails to converge.
- Weierstrass and Blancmange functions are continuous everywhere but differentiable nowhere.
- Koch snowflake is a fractal curve, not a function.
- Give an example of a function whose limit exists but which is not continuous.
- Give an example of a function that is continuous but not differentiable.
- Why does |x| fail to be differentiable at x = 0?
- Differentiate between a corner and a cusp.
- What is meant by a vertical tangent?
- Explain why x sin(1/x) is continuous but not differentiable at x = 0.
- State the hierarchy relating limits, continuity and differentiability.
- Why is the Koch snowflake not considered a function?
- Name two famous functions that are continuous everywhere but differentiable nowhere.
- Give one real-life application where differentiability is important.
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