Primary Tool
2D Graphing
The graph makes the distinction immediate. The parabola is both continuous and differentiable, while is continuous but has a corner at the origin where no single tangent slope exists.
One implication works, the other does not
If a function is differentiable at a point, then it is automatically continuous there. The derivative gives such strong local control that the function value cannot jump or tear.
But continuity alone only says the graph is unbroken. It does not say the graph has a single well-defined slope at that point.
What the corner in shows
The function is continuous at because the left-hand and right-hand values both meet at the same point. Nothing breaks in the graph.
Yet the slopes coming from the two sides disagree: the left-hand slope is and the right-hand slope is . Because those two tangent directions do not match, the derivative at does not exist.
What differentiability adds
Differentiability says more than 'the graph meets up.' It says the function has a good local linear approximation at the point.
That is why smooth-looking curves like are differentiable at the origin while cornered graphs like are not. Continuity is about staying connected; differentiability is about having one coherent local slope.
Common Pitfall
A graph that is unbroken is not automatically differentiable. Corners, cusps, and vertical tangents can all preserve continuity while destroying the derivative.
Try a Variation
Graph and compare it with . Is the graph continuous at ? Does it have a finite tangent slope there?
Related Pages
Keep moving through the cluster
Proof that a differentiable function is continuous
This proof shows why differentiability is a stronger condition than continuity. The difference factors into a difference quotient times , and that product is forced to zero.
Open proof →
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.
Open worked example →
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 →