Continuity and Differentiability

The whole of science is nothing more than a refinement of everyday thinking. — Albert Einstein

5.1 Introduction

This chapter continues differentiation from Class XI: continuity, differentiability, inverse trigonometric derivatives, and exponential / logarithmic functions.

5.2 Continuity

$f$ is continuous at $c$ if

$\lim_{x\to c} f(x)=f(c)$

That is: left-hand limit, right-hand limit, and $f(c)$ all exist and are equal.

$f$ is continuous if it is continuous at every point of its domain.

On $[a,b]$, continuity at the endpoints uses one-sided limits:

$\lim_{x\to a^+} f(x)=f(a),\qquad \lim_{x\to b^-} f(x)=f(b)$

Sum, difference, product, and quotient of continuous functions are continuous (quotient wherever the denominator is nonzero).

Polynomials, $\sin x$, $\cos x$, $e^x$, $\log x$ (on $(0,\infty)$), and $|x|$ are continuous on their usual domains.

5.3 Differentiability

$f$ is differentiable at $c$ if

$f'(c)=\lim_{h\to 0}\frac{f(c+h)-f(c)}{h}$

exists.

Every differentiable function is continuous. The converse is not true. (Classic example: $|x|$ at $0$.)

Chain rule

If $f=v\circ u$, $t=u(x)$, and both $\dfrac{dt}{dx}$ and $\dfrac{dv}{dt}$ exist, then

$\frac{df}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}$

Standard derivatives