Continuity

• A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.
• A function f(x) is said to be continuous at a point x = a, if \lim_{x \rightarrow a}=f(a)
• A function f(x) is said to be continuous at a point x = a, if LHL = RHL = f(a)
                  \Rightarrow \lim_{x \rightarrow a^{-}}f(x)=\lim_{x \rightarrow a^{+}}f(x)=f(a)
                  \Rightarrow \lim_{h \rightarrow 0}f(a-h)=\lim_{h \rightarrow 0}f(a+h)=f(a)
• A function f(x) is said to be left continuous at x = a, if \Rightarrow \lim_{x \rightarrow a^{-}}f(x)=f(a) i.e., LHL = f(a)
• A function f(x) is said to be right continuous at x = a, if \Rightarrow \lim_{x \rightarrow a^{+}}f(x)=f(a) i.e., RHL = f(a)
• A function f(x) is said to be continuous on (a, b) if f(x) is continuous at every point in (a, b). i.e., ∃ an arbitrary point c ∈ (a, b), f(x) is continuous at x = c.
• f(x) is continuous on [a, b] if
     (i) fx) is Right continuous at x = a. i.e., \lim_{x \rightarrow a^{+}}f(x)=f(a)
     (ii) f(x) is left continuous at x = b i.e., \lim_{x \rightarrow b^{-}}f(x)=f(b)
     (iii) f(x) is continuous on (a, b)
• The graph of the curve is said be continuous function through out the domain, the graph has no holes (or) no gaps.
• Properties of continuous functions:
  → If f and g are two continuous functions, then f + g, f − g, fg, \frac{f}{g} are also continuous.
  → The composite function of two continuous functions is also continuous.

Tips for continuity of functions:

• A function f(x) is continuous on R, then |f(x)| is also continuous on R but converse need not be true.
• Every polynomial function is continuous on R.
• Every constant function is continuous on R.
• A function f(x) = [x] (integral part of x) is continuous all real numbers except integers.
• Every exponential function and logarithmic function is continuous on its domains.
• → sin x is continuous on R
  → cos x is continuous on R
• → tan x is continuous on R-\left\{(2n+1)\frac{\pi}{2}\right\}, n \ \epsilon \ z
  → sec x is continuous on R-\left\{(2n+1)\frac{\pi}{2}\right\}, n \ \epsilon \ z
• → cot x is continuous on R − {nπ}, n ∈ z
  → cosec x is continuous on R − {nπ}, n ∈ z
• A function f(x) is not continuous, then it is said to be discontinuous
• 
• Missing point discontinuity: \lim_{x \rightarrow a} \ f(x) exists but f(a) does not exists.
    \therefore \lim_{x \rightarrow a} \ f(x) \neq f(a)
• Isolated point discontinuity: \lim_{x \rightarrow a} \ f(x) exists but it is not equal to f(a) even f(a) exists also
  \therefore \lim_{x \rightarrow a} \ f(x) \neq f(a)
• Finite type: If both LHL and RHL exists finitely but both are not equal i.e., LHL ≠ RHL
• Infinite type: If atleast one of the limits LHL (or) RHL goes to ∞ i.e., either LHL = ∞ (or) RHL = ∞.
• Oscillatory type: Limit value is oscillating between two finite values but not getting a particular constant.
• Jump discontinuity: The non negative difference of both LHL and RHL is called Jump discontinuity i.e.,
          Jump = |LHL − RHL|
        If both LHL and RHL must be finite.