Primary Tool
Proof Builder
Proof Builder makes the factorization argument easy to follow line by line. The proof is short, but each step matters: express the increment as a product, take limits, and then read continuity off the result.
The core idea
Continuity at means as . Differentiability gives exactly the right expression to study that difference.
For nonzero , you can factor the increment as . One factor tends to and the other tends to , so their product tends to .
Why this makes differentiability stronger
Continuity only asks that the function values meet up at the point. Differentiability asks for a stable local slope as well.
That extra slope information is what forces continuity automatically. The converse fails because an unbroken graph can still have a corner and therefore no derivative.
Common Pitfall
You cannot reverse this theorem. A function may be continuous at a point and still fail to be differentiable there, as happens for at .
Try a Variation
Apply the same increment argument to a constant function. What happens to the difference quotient, and how does the proof simplify?
Related Pages
Keep moving through the cluster
Why continuity and differentiability are different concepts
Differentiability is stronger than continuity. A graph can be unbroken at a point and still fail to have a derivative there if the local shape has a corner, cusp, or vertical tangent.
Open explanation →
Differentiate a polynomial from first principles
This worked example shows how the derivative definition creates an algebra problem first and a limit problem second. The simplification step is the point of the exercise.
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What a derivative means geometrically
The derivative is the slope of the tangent line, but that phrase only becomes useful when you compare the curve to nearby secant lines and local linear approximations.
Open explanation →