Discuss the continuity of the function f at x - -4 where f is defined by fx - 1-2sin x- - 4x for x -4 -14 for x-4

f(x) = 1 √(2)sin xπ 4x =(√(2)4) 1√(2) sin xπ4 x limx →π/4(√(2)4) (sin 45^o sin xπ4 x) limx→π/4(√(2)4) 2sin(π4 x2)cos(x ...

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Continuity in an Interval

Discuss the c...

Question

Discuss the continuity of the function $f$ at $x = \frac{π}{4}$ where $f$ is defined by
$f (x) = \frac{1 - \sqrt{2} sin x}{π - 4 x}$ for $x \neq \frac{π}{4}$
$= \frac{1}{4}$ for $x \neq \frac{π}{4}$

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f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1 - \sqrt{2} sin x}{π - 4 x}, & x \neq \frac{π}{4} a, & x = \frac{π}{4} \end{matrix}

x = \frac{π}{4},

a

f (x) = \frac{1 - t a n x}{1 - \sqrt{2} s i n x}

x \neq \frac{π}{4}

x = \frac{π}{4}

f (\frac{π}{4})

f (x)

[0, π]

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x + a \sqrt{2} sin x & , 0 \leq x \leq π / 4 2 x cot x + b & , \frac{π}{4} < x \leq \frac{π}{2} a cos 2 x - b sin x & , \frac{π}{2} < x < π \end{matrix}

f

[0, π]

(\frac{π}{6}, \frac{π}{3})

f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{\sqrt{2} cos x - 1}{cot x - 1}, & x \neq \frac{π}{4} k, & x = \frac{π}{4} \end{matrix}

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin x - cos x}{x - \frac{p i}{4}} & x \neq \frac{π}{4} k & x = \frac{π}{4} \end{matrix}

f

x = \frac{π}{4}

k

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