Continuity and Differentiability EXPLAINED with Examples

Explore the knowledge in this video

Open video knowledge graph →

• Continuity ExplanationAt 0:21
  Why this connection?

  At an interior point $a$, check that $f(a)$ is defined, that $\lim_{x\to a}f(x)$ exists, and that this limit equals $f(a)$. The two-sided limit requires agreeing left and right limits. A missing value violates the first test. A jump violates the existence of a common two-sided limit. A removable discontinuity can have a limit that differs from the assigned function value.

• Continuity ExplanationAt 0:21
  Why this connection?

  At an interior point $a$, check that $f(a)$ is defined, that $\lim_{x\to a}f(x)$ exists, and that this limit equals $f(a)$. The two-sided limit requires agreeing left and right limits. A missing value violates the first test. A jump violates the existence of a common two-sided limit. A removable discontinuity can have a limit that differs from the assigned function value. Differentiability requires a finite derivative to exist at the point. A smooth example has a well-defined tangent slope, but continuity alone does not supply that slope. At a corner, the left and right slopes disagree; at a cusp, a finite derivative can also fail to exist. These examples explain why a continuous graph need not be differentiable. The informal 'no sharp points' picture is an intuition, not a general theorem guaranteeing differentiability.