Overview of Continuity and Differentiability

Let f(x) be a real function and a be a point in its domain.

A function f is continuous at x = a iff all three conditions hold:

• f(a) is defined
• \[\lim_{x\to a}f(x)\] exists
• \[\lim_{x\to a}f(x)\] = f(a)

\[\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)\]

(a) Left Hand Continuity at x = a

A function is left continuous at x = a if:

1. f(a) exists

2. \[\lim_{x\to a^-}f(x)\mathrm{~exists}\]

3. \[\lim_{x\to a^-}f(x)=f(a)\]

(b) Right Hand Continuity at x = a

A function is right continuous at x = a if:

1. f(a) exists

2. \[\lim_{x\to a^+}f(x)\mathrm{~exists}\]

3. \[\lim_{x\to a^+}f(x)=f(a)\]

c) Continuity at x = a

A function is continuous at x = a iff it is both left continuous and right continuous at x = a.

A function fails to be continuous at x = a if any one of the following occurs:

1. f(a) is not defined

2. \[\lim_{x\to a}f(x)\] does not exist

   • Either LHL or RHL does not exist

   • Or LHL ≠ RHL

3. \[\lim_{x\to a}f(x)\] exists but \[\lim_{x\to a}f(x)\] ≠ f(a)

For the open interval:

A function f is said to be continuous on an open interval (a, b) if it is continuous at every point in the interval. 

For a closed interval:

A function f is said to be continuous on the closed interval [a,b] iff:

1. f is continuous at every point of (a,b)

2. f is right continuous at a

   \[\lim_{x\to a^+}f(x)=f(a)\]

3. f is left continuous at b

   \[\lim_{x\to b^-}f(x)=f(b)\]

The set of all points where a function is continuous is called its domain of continuity.

A function f(x) is said to be differentiable at x = a if Rf'(a) and Lf'(a) both exist and are equal; it is said to be non-differentiable. 

Left Derivative at x = c:

\[\lim_{h\to0^-}\frac{f(c+h)-f(c)}{h}\]

Right Derivative at x = c

\[\lim_{h\to0^+}\frac{f(c+h)-f(c)}{h}\]

A function f is said to have a derivative at any point x if

\[f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]

1. Chain Rule:

If u = g(x) and y = f(u), then

\[\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\]

2. Composite Function Form:

If h(x) = f(g(x)), then

\[h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)\]

3. Power of a Function:

If y = [f(x)]ⁿ, then

\[\frac{dy}{dx}=n[f(x)]^{n-1}\cdot f^{\prime}(x)\]

4. Inverse Function Formula:

\[\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}},\quad\frac{dx}{dy}\neq0\]

\[\frac{dy}{dx}\cdot\frac{dx}{dy}=1\]

5. Derivative of Absolute Value Function:

For y=∣x∣,

\[\frac{d}{dx}(|x|)=\frac{x}{|x|},\quad x\neq0\]

6. Special Results:

\[\frac{d}{dx}(x)=1\]

\[\frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2},x\neq0\]

\[\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}},\mathrm{~}x>0\]

\[\frac{d}{dx}(\sqrt{ax+b})=\frac{a}{2\sqrt{ax+b}}\]

A. Trigonometric Functions

B. Inverse Trigonometric Functions

\[\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}\]

\[\frac{d}{dx}(xy)=x\frac{dy}{dx}+y\]

• \[a^{0}=1\]​

• \[a^x\cdot a^y=a^{x+y}\]​

• \[(a^x)^y=a^{xy}\]

• \[a^{-x}=\frac{1}{a^x}\]

\[\frac{d}{dx}(e^x)=e^x\]

\[\frac{d}{dx}(a^x)=a^x\log a,\quad a>0,a\neq1\]

\[\frac{d}{dx}(e^{f(x)})=e^{f(x)}\cdot f^{\prime}(x)\]

\[\frac{d}{dx}(a^{f(x)})=a^{f(x)}\log a\cdot f^{\prime}(x)\]

(i) Product of two functions

If y = uv then, \[\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\]

(i) Product of three functions

If y = uvw then \[\frac{dy}{dx}=uv\frac{dw}{dx}+uw\frac{dv}{dx}+vw\frac{du}{dx}\]

Quotient Rule:

If \[y=\frac{u}{v}\] then \[\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\]

Reciprocal Rule:

\[\frac{d}{dx}{\left(\frac{1}{f(x)}\right)}=-\frac{f^{\prime}(x)}{[f(x)]^2}\]

First derivative:

If x = f(t), y = ϕ(t) then \[\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\]

Second derivative:

\[\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}\]

For a 2×2 determinant:

\[F^{\prime}(x)=
\begin{vmatrix}
f_1^{\prime}(x) & f_2(x) \\
g_1^{\prime}(x) & g_2(x)
\end{vmatrix}+
\begin{vmatrix}
f_1(x) & f_2^{\prime}(x) \\
g_1(x) & g_2^{\prime}(x)
\end{vmatrix}\]

For a 3×3 determinant:

\[\mathrm{F^{\prime}}(x)=
\begin{vmatrix}
f_1^{\prime}(x) & f_2^{\prime}(x) & f_3^{\prime}(x) \\
g_1(x) & g_2(x) & g_3^{\prime}(x) \\
h_1(x) & h_2(x) & h_3(x)
\end{vmatrix}+
\begin{vmatrix}
f_1(x) & f_2(x) & f_3(x) \\
g_1^{\prime}(x) & g_2^{\prime}(x) & g_3^{\prime}(x) \\
h_1(x) & h_2(x) & h_3(x)
\end{vmatrix}+
\begin{vmatrix}
f_1(x) & f_2(x) & f_3(x) \\
g_1(x) & g_2(x) & g_3(x) \\
h_1^{\prime}(x) & h_2^{\prime}(x) & h_3^{\prime}(x)
\end{vmatrix}\]

If a function f(x) is

1. Continuous on [a,b]

2. Differentiable on (a,b)

3. f(a) = f(b)

Then there exists at least one c ∈ (a,b) such that f′(c) = 0​

f a function f(x) is

1. Continuous on [a,b]

2. Differentiable on (a,b)

Then there exists at least one c ∈ (a,b) such that

\[f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\]