Continuity

Definition of Continuity

Continuity at a Point

Function f(x) is continuous at x = c if three conditions hold: f(c) is defined, limit of f(x) as x→c exists, and limit equals f(c).

Mathematical Expression

limₓ→c f(x) = f(c)

Continuity on an Interval

f(x) continuous on interval I if continuous at every point in I including endpoints (one-sided limits at boundaries).

Types of Continuity

Pointwise Continuity

Continuity evaluated at individual points. Basic notion for function behavior.

Continuity on Intervals

Continuous over all points in an interval. Guarantees no breaks or gaps.

Right and Left Continuity

Right-continuous if limit from right equals function value; similarly for left-continuity.

Types of Discontinuities

Removable Discontinuity

Limit exists but f(c) is undefined or differs from limit; fixable by redefining f(c).

Jump Discontinuity

Left and right limits exist but are not equal; function "jumps" at c.

Infinite Discontinuity

Limit approaches infinity or negative infinity at c; vertical asymptote present.

Oscillatory Discontinuity

Limit does not exist due to oscillation near c; no settled value.

Epsilon-Delta Definition

Formal Definition

f is continuous at c if ∀ε>0 ∃δ>0 s.t. |x−c|<δ → |f(x)−f(c)|<ε.

Interpretation

Arbitrarily close x-values yield arbitrarily close f(x)-values near c.

Use in Proofs

Foundation for rigorous continuity proofs and limit calculations.

∀ε>0, ∃δ>0: |x−c|<δ ⇒ |f(x)−f(c)|<ε

Properties of Continuous Functions

Algebraic Operations

Sum, difference, product, and quotient (denominator ≠ 0) of continuous functions remain continuous.

Composition

Composition of continuous functions is continuous at corresponding points.

Intermediate Value Property

Continuous functions attain all intermediate values between f(a) and f(b).

Boundedness

Continuous functions on closed intervals are bounded and attain extrema (Weierstrass theorem).

Intermediate Value Theorem

Theorem Statement

If f continuous on [a, b], and k between f(a) and f(b), ∃c∈(a,b) with f(c)=k.

Implications

Guarantees roots and intermediate values; foundation for root-finding algorithms.

Limitations

Requires continuity on closed interval; fails if discontinuous anywhere in [a,b].

Uniform Continuity

Definition

f continuous on domain D uniformly if ∀ε>0 ∃δ>0 s.t. ∀x,y∈D, |x−y|<δ → |f(x)−f(y)|<ε.

Difference from Pointwise

δ independent of point choice; stronger than ordinary continuity.

Examples

Continuous functions on closed intervals are uniformly continuous; f(x)=1/x on (0,1) is not.

Continuity in Practice

Checking Continuity

Evaluate limits and function values; verify equality at points of interest.

Using Limits

Calculate left and right limits for piecewise or boundary points.

Software Tools

Symbolic calculators and graphing software assist in continuity analysis and visualization.

Continuity of Piecewise Functions

Definition

Functions defined by multiple expressions over distinct intervals.

Continuity Conditions

Matching left and right limits and function values at boundary points.

Common Issues

Discontinuities often occur at piece boundaries; require careful limit evaluation.

Continuity and Limits

Limit Existence

Continuity at c requires limit of f(x) as x→c exists.

Relation to Discontinuities

Discontinuities correspond to limit failure or mismatch with f(c).

Limit Laws

Used to verify continuity via algebraic manipulation and limit evaluation.

Graphical Interpretation

Visual Continuity

Graph without breaks, jumps, or holes at point c indicates continuity.

Discontinuity Types

Removable holes, jumps, and vertical asymptotes visible graphically.

Practical Graphing

Sketching helps identify continuity and approximate limit behavior.

Applications of Continuity

Root Finding

Intermediate Value Theorem underpins bisection and other root algorithms.

Optimization

Continuous functions guarantee existence of maxima/minima on closed intervals.

Engineering and Physics

Modeling physical phenomena assumes continuity for realistic behavior.

Further Calculus

Continuity prerequisite for differentiability and integration.

References

• Apostol, T.M., Calculus, Volume I, Wiley, 1967, pp. 45-75.
• Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1976, pp. 100-130.
• Stewart, J., Calculus: Early Transcendentals, Brooks Cole, 2015, pp. 150-185.
• Spivak, M., Calculus, Publish or Perish, 1994, pp. 80-110.
• Thomas, G.B., Calculus and Analytic Geometry, Addison-Wesley, 2002, pp. 120-160.