continuity-and-differentiability

Important Points

• A function is said to be continuous at x = a if: ${\lim }_{x\to a^{+}}f(x)={\lim }_{x\to a^{-}}f(x)=f(a)$
• We can do the following if and only if f(x) is continuous ${\lim }_{x\to a}f(g(x))=f({\lim }_{x\to a}g(x))$
• Any number x can be represented by $x=[x]+\{x\}$ where [] is greatest integer function and {} is fractional part function
• Wheneber greatest integer function is causing a problem in any question, break it into it’s normal and fractional part
• fractional part function lies between 0 and 1
  • It can be equal to zero
  • It cannot be equalt to one
• If f(x) is a function that takes a jump at x = a and g(x) is a function that is joined at x = a and zero at x=a then product of these functions will be joined(continuous) and zero at x = a
• In order to find discontinuity, you just have to find the points where the graph of the given function takes a jump i.e gives a completely different and unrelated value than rest or most of the function.
• for the following format of a question $f(x)=\{g(x);x\in \mathbb{Q},h(x);x\notin \mathbb{Q}\}$ this function will be continuous only for the points where h(x) = g(x)
• While using expansions of functions, only expand the function untill the related power: if x² is given, expand your function only till x² or 2nd value
• Let’s say you have a parabollic function f(x) which has it’s highest point at y = b and lowest at y = a then $y=[f(x)]$ is discontinuous at points $n=2(b-a)$
• If g(x) is continuous at x=a and f(x) is continuous at x=g(x) then f(g(x)) is continuous at x=a
• If g(x) is discontinuous at x=a, then f(g(x)) may or may not be discontinuous at x=a
• If g(x) is undefined at x=a, then f(g(x)) is discontinuous at x=a
• In order to find points of discontinuity for f(g(x)) find the points where g(x) and f(g(x)) is undefined.
• f(x) is continuous in (a,b) if f(x) is continuous for every $x\in (a,b)$
• f(x) is continuous in [a,b] if
  • f(x) is continuous for every $x\in (a,b)$
  • RHL|_a = f(a)
  • LHL|_b = f(b)

Types of discontinuity

1. Removable
   • Isolated point discontinuity - When f(a) exists but
     • LHL|_a = RHL|_a != f(a)
   • Missing point discontinuity - When f(a) does not exists but
     • LHL|_a = RHL|_a
2. Irremovable
   • Discontinuity of 1st kind - When both LHL|_a and RHL|_a exists but LHL|_a != RHL|_a
   • Discontinuity of 2nd kind - When either LHL|_a or RHL|_a do not exist

Differentiability

• For any function, f(x), it’s right and left hand derivative are defined as $RHD{|}_a={\lim }_{x\to a^{+}}\frac{f(x+h)-f(x)}{h}$ $LHD{|}_a={\lim }_{x\to a^{-}}\frac{f(x-h)-f(x)}{-h}$

• A function f(x) is said to be differentiable at x=a if $LHD{|}_a=RHD{|}_a=\text{finite}$

• A differentiable function is always continuous

• A continuous function may or may not be differentiable

• A non-differentiable function may or may not be continuous

• A discontinuous function is always non-differentiable

• If $LHL{|}_a$ and $RHL{|}_a$ are finite then f(x) is continuous at x=a even if $LHL{|}_a\ne RHL{|}_a$

• Following function is differentiable at x=a $(x-a)|x-a|$

• For following function $f(x)=\{(x-n{)}^P\sin (\frac{1}{x-n});x\ne n0;x=n\}$ at x=n

  • if P < 1: it is not continuous
  • if P = 1: it is continuous but not differentiable
  • if P > 1: it is differentiable
  • if P > 2: $\frac{df(x)}{dx}$ is continuous but not differentiable

• If f(x) is differentiable at x=a and g(x) is not differentiable at x=a then $f(x)\pm g(x)$ is always non-differentiable at x=a. Nothing can be said about their product or division

• If both f(x) and g(x) are differentiable at x=a then their binary operations are always differentiable

• If both f(x) and g(x) are non differentiable at x=a then nothing can be said about their binary operations.

• f(x) is said to be differentiable in (a,b) if it is differentiable for every x $\in$ (a,b)

• f(x) is said to be differentiable in [a,b]

  • if it is differentiable for every x $\in$ (a,b)
  • $RHD{|}_a$ exists at x=a
  • $LHD{|}_b$ exists at x=a