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Differential Calculus

Q.Can we say that all even functions are continuous and non differentiable at x=0 and all odd functions are continuous and differentiable at x=0:?

siddharth gupta

12 Years agoGrade

3 Answers

Arun Kumar

12 Years ago

Hello Student,

No we can’t generally say that.
In even function take the example of cosx
its continous and differentiable at 0.
In odd function take example of sin(1/x).
its non continous and non differentiable at 0.

Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty

Jitender Singh

12 Years ago

Ans: Yes

Sol:

Let f(x) be the even function, We have

$f(x)= f(-x)$

$f(0)= 0$

Proof for continuity:

Right Hand Limit (RHL) = f(0) = Left Hand Limit (LHL)

$\lim_{h\rightarrow 0}f(h) = 0 = \lim_{h\rightarrow 0}f(-h)$

$RHL = \lim_{h\rightarrow 0}f(h) = f(0) = 0$

$LHL = \lim_{h\rightarrow 0}f(-h) = f(0) = 0$

Proof for Non Differentiability:

For f to be differentiable,

Left Hand Derivative (LHD) = Right Hand Derivative (RHD)

$\lim_{h\rightarrow 0}\frac{f(-h)-f(0)}{-h} = \lim_{h\rightarrow 0}\frac{f(h)-f(0)}{h}$

$LHD = \lim_{h\rightarrow 0}\frac{f(-h)-f(0)}{-h} = \lim_{h\rightarrow 0}-\frac{f(h)}{h}$

$RHD = \lim_{h\rightarrow 0}\frac{f(h)-f(0)}{h} = \lim_{h\rightarrow 0}\frac{f(h)}{h}$

$LHD \neq RHD$

Let f(x) be odd,

$f(x)= -f(-x)$

$f(0)= 0$

Proof for continuity:

RHL = f(0) = LHL

$RHL = \lim_{h\rightarrow 0}f(h) = f(0) = 0$

$LHL = \lim_{h\rightarrow 0}f(-h) = f(0) = 0$

Proof for Differentiability:

LHD = RHD

$LHD = \lim_{h\rightarrow 0}\frac{f(-h)-f(0)}{-h} = \frac{-f(h)}{-h} = \frac{f(h)}{h}$

$RHD = \lim_{h\rightarrow 0}\frac{f(h)-f(0)}{h} = \frac{f(h)}h}$

Thanks & Regards

Jitender Singh

IIT Delhi

askIITians Faculty

Jitender Singh

12 Years ago

There is a little mistake in the previous solution.

In case of even, differentiability depend on the f(x). For example,

f(x) = x², LHD = RHD = 0

f(x) = |x|, LHD = -1 RHD = 1

x² is differentiable but |x| is not.

Similarly for the odd function. For example,

f(x) = x³, LHD = 0 RHD = 0 Differentiable

f(x) = x^1/3, Derivative becomes infinite. So non differentiable

Thanks & Regards

Jitender Singh

IIT Delhi

askIITians Faculty

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