Continuity and Rules of Differentiation

Continuity of a Function

A function \(y=f(x)\) is said to be continuous at \(a\) if \(\lim_{x \rightarrow a} f(x)\) exists and \[\lim_{x \rightarrow a} f(x) = f(a)\]

Discontinuity: Example

Differentiability and Continuity

\(f'(x_0)\) exists if the following limit exists: \[f'(x_0)= \lim_{x \rightarrow x_0} \frac{f(x)-f(x_{0})}{x-x_0}\]

A function \(y=f(x)\) is continuous at \(x_0\) if \[\lim_{x \rightarrow x_0} f(x) = f(x_0)\]

Any connection?

Differentiability and Continuity

Continuity is a necessary condition for differentiability, but it is not sufficient.

\(f\) is not continuous \(\Longrightarrow\) \(f\) is not differentiable

\(f\) is continuous \(\Longrightarrow\) \(f\) could be differentiable or not

Continuous but Not Differentiable

So How to Differentiate Functions?

Rules of differentiation, easier than taking the limit each time

Constant function rule: for \(f(x)=k\), \(f'(x)=0\)

Power function rule: for \(f(x)=x^{n}\), \(f'(x)=n x^{n-1}\)

Generalized power function rule: for \(f(x)=c x^{n}\), \(f'(x)=c n x^{n-1}\)

Rules of Differentiation

Two or more functions of one variable

Sum-difference rule \[\frac{d}{d x}[f(x) \pm g(x)]=f'(x) \pm g'(x)\]

Product rule \[\frac{d}{d x}[f(x) g(x)]=f(x) g'(x)+f'(x) g(x)\]

Rules of Differentiation

Two or more functions of one variable

Quotient rule \[\frac{d}{d x} \frac{f(x)}{g(x)}=\frac{f'(x) g(x)-f(x) g'(x)}{g(x)^2}\]

Inverse function rule \[\frac{d x}{d y}=\frac{1}{d y / d x}\]

Rules of Differentiation

Functions of different variables

Chain rule: for \(z=f(y)\), \(y=g(x)\) \[\frac{d z}{d x}=\frac{d z}{d y} \cdot \frac{d y}{d x}=f'(y) g'(x)\]

Homework Questions

• Exercise 7.1: 3

• Exercise 7.2: 3 (d), (e), 7, 8

• Exercise 7.3: 1-6

Textbook reference: 6.7, 7.1-7.3