Mathematics Revision Of CONTINUITY AND DIFFERENTIABILITY FOR CBSE-NCERT
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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"}.

### Differentiability

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 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]

### 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.