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Introduction to Nervous System

Why different...

Question

Why differentiability is defined or we can say checked in open interval while continuity in closed interval?

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f (x) = 1 - x^{2} + x^{3}

is defined in the closed interval

[- 1, 1]

x

, in the open interval

(- 1, 1)

for which the mean value theorem is satifsied, is

Q. Consider the function

y = | x - 1 | + | x - 2 |

in the interval

[0, 3]

and discuss the continuity and differentiability of the function in this interval.

f

be a differentiable function on the open interval

(a, b)

. Which of the following statements must be true?

(i) f

is continuous on the closed interval

(a, b)

(i i) f

is bounded on the open interval

(a, b)

(i i i)

a < a_{1} < b_{1} < b

f (a_{1}) < 0 < f (b_{1})

, then there is a number

c

a_{1} < c < b_{1}

f (c) = 0

Q. The function

f

is defined, where

x = 2 t - | t |, t \in R

. Draw the graph of

f

for the interval

- 1 \leq x \leq 1

. Also discuss its continuity and differentiability at

x = 0

Q. Discuss the continuity and differentiability of the

f (x) = |x| + |x - 1| in the interval (- 1, 2)

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