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  • Confused! kindly explain, - Letbe a function defined as then, is: - Limit , continuity and differentiability - JEE Main

 

Let f: \mathbb R\rightarrow \mathbb R be a function defined as

f(x) = \left\{\begin{matrix} \;\;5,\;if\; x\leq 1\\ a + bx, \; if\; 3\leq x\leq 5\\ 30,\;if\;x\geq 5\\ \end{matrix}\right.

then, f is:

  • Option 1)

    continuous if a = 5 and b = 5

  • Option 2)

    continuous if a = 10 and b = 10

  • Option 3)

    continuous if a = 0 and b = 5 

  • Option 4)

    Not continuous for any values of a and b

Answers (1)

best_answer

 

Continuity at a point -

A function f(x)  is said to be continuous at  x = a in its domain if 

1.  f(a) is defined  : at  x = a.

2. \lim_{x\rightarrow a}\:f(x)\:exists\:means\:limit\:x\rightarrow a

of  f(x) at  x = a exists from left and right.

3. \lim_{x\rightarrow a}\:f(x)=f(a)\:then\:the\:limit\:equals \:the\:value\:at\:x=a

-

 

f(x)=\left\{\begin{matrix} 5 & if & x\leq 1\\ a+bx& if &1<x<3 \\ b+5x&if &3\leq x< 5 \\ 30 & if& x\geqslant 5 \end{matrix}\right.

Now, 

f(1)=5,\; \; \; \; f(1^{-})=5\: \: \: \: and\: \: \: \: f(1^{+})=a+b

f(3^{-})=a+3b,\; \; \; \; f(3)=b+15\: \: \: \: and\: \: \: \: f(3^{+})=b\pm 5

f(5^{-})=b+25,\; \; \; \; f(5)=30\: \: \: \: and\: \: \: \: f(5^{+})=30

from above 

f(x) is not continuous for any value of a & b  


Option 1)

continuous if a = 5 and b = 5

Option 2)

continuous if a = 10 and b = 10

Option 3)

continuous if a = 0 and b = 5 

Option 4)

Not continuous for any values of a and b
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