Continuity Conditions

Understanding continuity is key in calculus, as it helps us analyze functions and their behavior. This includes knowing when a function is continuous at a point, on an interval, and how continuity relates to limits and differentiability.

1. Definition of continuity at a point

   • A function ( f(x) ) is continuous at a point ( c ) if:
     • ( f(c) ) is defined.
     • The limit of ( f(x) ) as ( x ) approaches ( c ) exists.
     • The limit equals the function value: ( \lim_{x \to c} f(x) = f(c) ).

2. Left-hand and right-hand continuity

   • Left-hand limit: ( \lim_{x \to c^-} f(x) ) must equal ( f(c) ) for left-hand continuity.
   • Right-hand limit: ( \lim_{x \to c^+} f(x) ) must equal ( f(c) ) for right-hand continuity.
   • A function is continuous at ( c ) if both left-hand and right-hand limits exist and are equal to ( f(c) ).

3. Continuity on an interval

   • A function is continuous on an interval if it is continuous at every point within that interval.
   • Intervals can be open, closed, or half-open.
   • Continuous functions on closed intervals are particularly important for applying the Extreme Value Theorem.

4. Intermediate Value Theorem

   • If ( f ) is continuous on the interval ([a, b]) and ( N ) is any value between ( f(a) ) and ( f(b) ), then there exists at least one ( c ) in ((a, b)) such that ( f(c) = N ).
   • This theorem guarantees the existence of solutions within an interval.

5. Continuous function properties

   • Continuous functions can be added, subtracted, multiplied, and divided (except by zero) to produce new continuous functions.
   • The composition of continuous functions is also continuous.
   • Continuous functions on closed intervals are bounded and attain their maximum and minimum values.

6. Types of discontinuities (removable, jump, infinite)

   • Removable discontinuity: A hole in the graph where the limit exists but does not equal the function value.
   • Jump discontinuity: The left-hand and right-hand limits exist but are not equal, causing a "jump" in the graph.
   • Infinite discontinuity: The function approaches infinity or negative infinity as ( x ) approaches a certain value.

7. Continuity of composite functions

   • If ( f ) is continuous at ( g(c) ) and ( g ) is continuous at ( c ), then the composite function ( f(g(x)) ) is continuous at ( c ).
   • This property is crucial for analyzing more complex functions built from simpler continuous functions.

8. Continuity of elementary functions

   • Polynomial functions, rational functions (where the denominator is not zero), trigonometric functions, exponential functions, and logarithmic functions are all continuous on their respective domains.
   • Understanding the continuity of these functions helps in solving limits and integrals.

9. Continuity and differentiability relationship

   • If a function is differentiable at a point, it is also continuous at that point.
   • However, continuity does not imply differentiability; a function can be continuous but not have a derivative at certain points (e.g., sharp corners).

10. Extreme Value Theorem

    • If a function is continuous on a closed interval ([a, b]), then it attains both a maximum and a minimum value at least once within that interval.
    • This theorem is essential for optimization problems in calculus.