f (x) = |x - 3| =
Let c be a real number.
Case I: c < 3. Then f(c) = 3 – c.
$\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) = $\underset{\mathit{x}\to \mathit{c}}{\lim }$ (3 - x) = 3 - x
Since, $\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) = f (c), f is continuous at all negative real numbers.
Case II: c = 3. Then f(c) = 3 – 3 = 0
$\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) = $\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) (x - 3) = 3 - 3 = 0
Since, $\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) f (x) = f (3), f is continuous at x = 3.
Case III: c > 3. Then f(c) = c – 3.
$\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) = $\underset{\mathit{x}\to \mathit{c}}{\lim }$ (x - 3) = x - 3
Since, $\underset{\mathit{x}\to \mathit{c}}{\lim }$ f (x) = f (c), f is continuous at all positive real numbers
Therefore, f is continuous function.
Now, we need to show that f(x) = |x - 3|, x ∊ R is not differentiable at x = 3.
Consider the left hand limit of f at x = 3
$\underset{\mathit{h}\to 0^{-}}{\lim }$ $\frac{\mathit{f}\left(3+\mathit{h}\right)-\mathit{f}\left(3\right)}{\mathit{h}}$ = $\underset{\mathit{h}\to 0^{-}}{\lim }$ $\frac{\left|3+\mathit{h}-3\right|-|3-3|}{\mathit{h}}$ = $\underset{\mathit{h}\to 0^{-}}{\lim }$ $\frac{\left|\mathit{h}\right|-0}{\mathit{h}}$ = $\underset{\mathit{h}\to 0^{-}}{\lim }$ $\frac{-\mathit{h}}{\mathit{h}}$ = - 1
h < 0 ⇒ |h| = - h
Consider the right hand limit of f at x = 3
$\underset{\mathit{h}\to 0^{+}}{\lim }$ $\frac{\mathit{f}\left(3+\mathit{h}\right)-\mathit{f}\left(3\right)}{\mathit{h}}$ $\underset{\mathit{h}\to 0^{+}}{\lim }$ $\frac{\left|3+\mathit{h}-3\right|-|3-3|}{\mathit{h}}$ = $\underset{\mathit{h}\to 0^{+}}{\lim }$ $\frac{\left|\mathit{h}\right|-0}{\mathit{h}}$ = $\underset{\mathit{h}\to 0^{+}}{\lim }$ $\frac{\mathit{h}}{\mathit{h}}$ = 1
h > 0 ⇒ |h| = h
Since the left and right hand limits are not equal, f is not differentiable at x = 3.
OR
y = a $\left(\mathit{c}\mathit{o}\mathit{s}\mathit{t}+\mathit{l}\mathit{o}\mathit{g}\mathit{t}\mathit{a}\mathit{n}\frac{\mathit{t}}{2}\right)$ find $\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{t}}$ = a $\left|\frac{\mathit{d}}{\mathit{d}\mathit{t}}+\mathit{c}\mathit{o}\mathit{s}\mathit{t}+\frac{\mathit{d}}{\mathit{d}\mathit{t}}\left(\mathit{l}\mathit{o}\mathit{g}\mathit{t}\mathit{a}\mathit{n}\frac{\mathit{t}}{2}\right)\right|$
= a $\left|-\mathit{s}\mathit{i}\mathit{n}\mathit{t}+\mathit{c}\mathit{o}\mathit{t}\frac{\mathit{t}}{2}\times \mathit{s}\mathit{e}{\mathit{c}}^3\frac{\mathit{t}}{2}\times \frac{1}{2}\right|$
= a $\left|-\mathit{s}\mathit{i}\mathit{n}\mathit{t}+\frac{1}{2\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{t}}{2}\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{t}}{2}}\right|$
= a $\left(-\mathit{s}\mathit{i}\mathit{n}\mathit{t}+\frac{1}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}\right)$ = a $\left(\frac{-\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{t}+1}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}\right)$ = a $\frac{\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{t}}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}$
x = a sin t
$\frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{t}}$ = a $\frac{\mathit{d}}{\mathit{d}\mathit{t}}$ sin t = a cos t
∴ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\left(\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)}{\left(\frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{t}}\right)}$ = $\frac{\left(\mathit{a}\frac{\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{t}}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}\right)}{\mathit{a}\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$ = $\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{t}}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}$ = cot t
$\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ = - $\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\mathit{t}\frac{\mathit{d}\mathit{t}}{\mathit{d}\mathit{x}}$ = - $\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\mathit{t}\times \frac{1}{\mathit{a}\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$ = - $\frac{1}{\mathit{a}\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{t}\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$