Differentiability Implies Continuity

Definition

In the context of calculus, differentiability is a “stronger” property than continuity.

Why It Matters

In science and engineering, “smoothness” is a high-level safety guarantee. If you know a system is differentiable, you are guaranteed it won’t have sudden, unpredictable jumps (discontinuities). This theorem allows us to build stable control systems and predictive models, knowing that as long as we can calculate a rate of change, the underlying process is fundamentally connected and unbroken.

Core Concepts

• Theorem If a function $f$ has a derivative at $x = c$, then $f$ must be continuous at $x = c$.

  • How to read: “If the derivative f prime of c exists, then the function f is continuous at c.”
  • Meaning: Smoothness (a defined tangent) is a stronger condition than no gaps; differentiable always implies continuous, never the reverse.

• Proof Sketch If $f'(c)$ exists, then: $\lim_{x \to c} [f(x) - f(c)] = \lim_{x \to c} \left[ \frac{f(x) - f(c)}{x - c} \cdot (x - c) \right] = f'(c) \cdot 0 = 0$

  • How to read: “The limit as x approaches c of the quantity f of x minus f of c equals the limit of the difference quotient multiplied by the quantity x minus c, which equals f prime of c times zero, resulting in zero.”
  • Meaning: Rewrite the function change as (slope) × (run); the slope is finite but the run shrinks to zero, forcing the rise to zero—hence continuity.

This implies $\lim_{x \to c} f(x) = f(c)$, which is the definition of continuity.

• How to read: “The limit of f of x as x approaches c equals f of c.”

• Meaning: The output value at c matches what nearby inputs approach—no jump at the point.

• Caution The converse is false. A function can be continuous at a point without being differentiable there. A prime example is $f(x) = |x|$, which is continuous at $x = 0$ but has a “corner” and thus no derivative there.

  • How to read: “The function f of x equals the absolute value of x is continuous at zero but is not differentiable at that point.”
  • Meaning: Continuity only rules out holes and jumps; corners can still block a unique tangent slope.

Connected Concepts

• Continuity at a Point
• Failure of Differentiability
• Multivariable Limits
• Multivariable Continuity