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continuity at a point

`\color{green} ✍️` If the left hand limit, right hand limit and the value of the function at `x = c` exist and equal to each other, then f is said to be continuous at `x = c`.

`color{orange}{lim_(x->c^-) f(x) =f(c) = lim_(x->c^+) f(x)}`

continuity at an interval

`\color{green} ✍️` Suppose `f` is a function defined on a closed interval `[a, b]`, then for `f` to be continuous, it needs to be continuous at every point in `[a, b]` including the end points `a` and `b`.

`=>` Continuity of `f` at a means `color{green}{lim_(x-> a^+) f(x) =f(a)}`

`=>` and continuity of `f` at `b` means `color{green}{lim_(x-> b^(-)) f(x) =f(b)}`

Algebra of continuous functions :

●`color{green} ✍️: ` Suppose `f` and `g` be two real functions continuous at a real number `c`.

(1) `color{red}{f + g}` is continuous at `x = c`.

(2) `color{red}{f - g}` is continuous at `x = c`.

(3) `color{red}{f * g}` is continuous at `x = c`.

(4) `color{red}(f/g)` is continuous at `x = c`, (provided `g (c) ≠ 0`).

●`color{red}{" If f and g are real valued functions such that" \ \ (f o g) \ \ "is defined at c."}`

`color{red}{" If g is continuous at c and if f is continuous at g (c) , then (f o g) is continuous at c"}`.


Theorem : If A function f is differentiable at a point c, then it is also continuous at that point.

`color{red}{"Note : So Every differentiable function is continuous at "}`
`color{red}{" that point but that doesn't mean Every continuous "}`
`color{red}{"function is differentiable at the same point."}`

The following rules of algebra of derivatives :

(1) `(u pm v)' = u' pm v'`

(2) `(uv)'= u'v + uv'` (Leibnitz or product rule)

(3) `(u/v)' =((u'v -uv')/v^2)` wherever `v != 0` (Quotient rule).

Differentiation formulas list :

Chain Rule Of Derivatives

`\color{green} ✍️` (i) If `f` be a real valued function which is a composite of two functions `u` and `v` ; i.e.,` f = v o u`. Suppose `t = u(x)` and if both `(dt)/(dx)` and `(dv)/(dt)` exist,

we have `color{orange}{(df)/(dx) =(dv)/(dt) * (dt)/(dx)}`

Derivatives of implicit functions

Derivatives of parametric functions

`\color{green} ✍️` A relation expressed between two variables `x` and `y` in the form `x = f(t), y = g (t)` is said to be parametric form with `t` as a parameter.

`\color{green} ✍️` In order to find derivative of function in such form, we have by chain rule.

`color{red}{(dy)/(dt) =(dy)/(dx) * (dx)/(dt)}`

or `(dy)/(dx) = ((dy)/(dt))/((dx)/(dt)) ` (whenever `(dx)/(dt) ne 0`)

Thus `=> (dy)/(dx) = (g'(t))/(f'(t)) ` (as `(dy)/(dt) = g'(t)` and `(dx)/(dt) = f'(t)`) [provided `f ′(t) ≠ 0`]

Logarithmic and exponential derivatives

Second Order Derivative

`\color{green} ✍️` Let `y = f (x)`. Then `(dy)/(dx)= f'(x)`.....................(1)

`\color{green} ✍️` If `f ′(x)` is differentiable, we may differentiate (1) again w.r.t. `x`. Then, the left hand side becomes `color{red}{d/(dx) ((dy)/(dx))}` which is called the second order derivative of `y` w.r.t. `x` and is denoted by `(d^2y)/(dx^2)`

`\color{green} ✍️` The second order derivative of `f (x)` is denoted by `f ″(x)`. It's denoted by `D^2 y ` or `y″` or `y^2` if `y = f (x)`. We remark that higher order derivatives may be defined similarly.

rolle's theorem

Mean value theorem