Kết nối Khả vi và Liên tục: Khi nào Đạo hàm Tồn tại và Không tồn tại

Smooth implies connected — but not the reverse

• If a function has a derivative at a point, its graph is not just unbroken there — it is smooth.
• Differentiability 可微性 at a point is a stronger condition than continuity.
• Key fact: differentiable $\Rightarrow$ continuous. A tangent line can't exist at a break.
• But the reverse fails: a continuous graph can still have a sharp spot with no derivative.

Why differentiable forces continuous

• A derivative is the limit of secant slopes; for that limit to be finite, the function can't jump.
• If $f$ jumped at $c$, the secant slopes near $c$ would blow up — no finite tangent slope.
• So having $f'(c)$ guarantees $f$ is continuous 连续 at $c$.
• Contrapositive (handy on exams): if $f$ is not continuous at $c$, it is not differentiable there.

Continuous but not differentiable: a corner 尖角 and a vertical tangent 竖直切线

Where the derivative fails to exist

• Continuity is not enough — three shapes break differentiability even with no gap:
• A corner: the left and right slopes differ (like $|x|$ at $0$).
• A cusp 尖点: the slopes shoot to $+\infty$ and $-\infty$ (like $x^{2/3}$ at $0$).
• A vertical tangent: the tangent is vertical, so its slope is undefined (like $\sqrt[3]{x}$ at $0$).

It all comes back to the tangent

• A derivative exists only when there is one well-defined tangent line with a finite slope.
• A corner has two different tangents; a cusp/vertical tangent has an infinite slope — neither gives a single finite value.
• $|x|$ is continuous everywhere but has no derivative at $0$ — the classic counterexample.
• So: check continuity first (necessary), then check for a single finite tangent slope (the rest).