Continuity & Differentiability Class 12 Notes Maths Chapter 5

• A function f(x) is said to be continuous at x= a if 
  lim_x→a^- f(x)= lim_x→a⁺ f(x)=f(a)
  Thus, unlike limits, for continuity it is essential for the function to be defined at that particular point and the limiting value of the function should be equal to f(a).
• The function f(x) will be discontinuous at x =a in either of the following situations:
  1. lim_x→a^- f(x) and lim_x→a⁺ f(x) exist but are not equal.
  2. lim_x→a^- f(x) and lim_x→a⁺ f(x) exist and are equal but not equal to f(a).
  3. f(a) is not defined.
  4. At least one of the limits does not exist.
• If you are required to comment on the continuity of a function, then you may just look for the points on the domain where the function is not defined. 

Some important properties of continuous functions:
If the functions f(x) and g(x) are both continuous at x =a then the following results hold true:
1. cf (x) is continuous at x =a where c is any constant.
2. f(x) + g(x) is continuous at x = a.
3. f(x).g(x) is continuous at x= a.
4. f(x)/g(x) is continuous at x= a, provided g(a) ≠ 0.

• If a function f is continuous in (a, b), it means it is continuous at every point of (a, b).
• If f is continuous in [a, b] then in addition to being continuous ay every point of domain, f should also be continuous at the end points i.e. f(x) is said to be continuous in the closed interval [a, b] if
  1. f(x) is continuous in (a, b)
  2. lim_x→a⁺ f(x)=f(a)
  lim_x→a^- f(x)=f(a)
• While solving problems on continuity, one need not calculate continuity at every point, in fact the elementary knowledge of the function should be used to search the points of discontinuity.
• In questions like this where a function h is defined as
  
  h(x) = f(x) for a < x < b
  g(x) for b < x < c
  The functions f and g are continuous in their respective intervals, then the continuity of function h should be checked only at the point x = b as this is the only possible point of discontinuity.
• If the point ‘a’ is finite, then the necessary and sufficient condition for the function f to be continuous at a is that lim_x→a^- f(x) and lim_n→a⁺ f(x) should exist and be equal to f(a).
• A function continuous on a closed interval [a, b] is necessarily bounded if both a and b are finite. This is not true in case of open interval.
• If the function u = f(x) is continuous at the point x=a, and the function y=g(u) is continuous at the point u = f(a), then the composite function y=(gof)(x)=g(f(x)) is continuous at the point x=a.
• Given below is the table of some common functions along with the intervals in which they are continuous:

  Functions f(x)	Interval in which f(x) is continuous
  Constant C	(-∞,∞)
  bn, n is an integer > 0	(-∞,∞)
  |x-a|	(-∞,∞)
  x-n, n is a positive integer.	(-∞,∞)-{0}
  a0xn + a1xn-1 +........ + an-1x + an	(-∞,∞)
  p(x)/q(x), p(x) and q(x) are polynomials in x	R - {x:q(x)=0}
  sin x	R
  cos x	R
  tan x	R-{nπ:n=0,±1,........}
  cot x	R-{(2n-1)π/2:n=0,±1,± 2,........}
  sec x	R-{(2n-1)π/2:n=0,±1,± 2,........}
  ex	R
  ln x	(0, ∞)

If you know the graph of a function, it can be easily judged without even solving whether a function is continuous or not. The graph below clearly shows that the function is discontinuous.

• If lim_x→a^- f(x) = L₁ and lim_x→a⁺ f(x) = L₂, where L₁ and L₂ are both finite numbers then it is called discontinuity of first kind or ordinary discontinuity.
  
  
• 
  
• 
  
• 
  
• 
  
• 
  
• 
  
• A function is said to have discontinuity of second kindif neither lim_x→a⁺ f(x) nor lim_x→a^- f(x) exist.
• If any one of lim_x→a⁺ f(x) or lim_x→a^- f(x) exists and the other does not then the function f is said to have mixed discontinuity.
• If lim_x→a f(x) exists but is not equal to f(a), then f(x) has removable discontinuity at x = a and it can be removed by redefining f(x) at x=a.
• If lim_x→a f(x) does not exist, then we can remove this discontinuity so that it becomes a non-removable or essential discontinuity.
• A function f(x) is said to have a jump discontinuity at a point x=a if, lim_x→a^-f(x) ≠ lim_x→a⁺ f(x) and f(x) and may be equal to either of previous limits.
  The concepts of limit and continuity are closely related. Whether a function is continuous or not can be determined by the limit of the function.