CONTINUITY

• Continuity on a point

  A function f(x) is continuous at x = a if:

  i. f(a) is defined

  ii. lim(x->a) f(x) exists

  iii. lim(x->a) f(x) = f(a)
• Polynomials are continuous everywhere
• Removable discontinuity


  A function f(x) has a removable discontinuity at x = a if:

  i. lim(x->a) f(x) exists and f(a) exists but lim(x->a) f(x) ≠ f(a)

  ii. lim(x->a) f(x) exists but f(a) does not exist
• Jump discontinuity


  A function f(x) has a jump discontinuity at x = a if lim(x->a+) f(x) and lim(x->a-) f(x) exist but are not equal. The size of the jump is |lim(x->a+) f(x) - lim(x->a-) f(x)|.
• Essential discontinuity


  A function f(x) at x = a if at least one of the one-sided limits does not exist or is infinite.
• Continuity on an interval

  i. f is continuous on (a,b) if f is continuous at every point in the open interval (a,b)

  ii. f is continuous on [a,b) if f(a) exists and f is continuous on (a,b)

  iii. f is continuous on (a,b] if f(b) exists and f is continuous on (a,b)

  iv. f is continuous on [a,b] if f is continuous on (a,b] and [a,b)