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# Determine if f defined by f(x) = {(x^{2 }sin1/x, if x≠0) (0, if x=0) is a continuous function?

**Solution:**

The given function is

f(x) = {(x^{2 }sin1/x, if x ≠ 0) (0, if x = 0)

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

If c ≠ 0, then f(c) = c^{2} sin 1/c

lim_{x→c} f(x) = lim_{x→c} (x^{2} sin1/x)

= (lim_{x→c} x^{2}) (lim_{x→c} sin1/x)

= c^{2}sin 1/c

⇒ lim_{x→c} f(x) = f(c)

Therefore, f is continuous at all points x, such that x ≠ 0.

Case II:

If c = 0, then f(0) = 0

lim_{x→0−} f(x) = lim_{x→0−} (x^{2 }sin1/x)

= lim_{x→0} (x^{2 }sin1/x)

It is known that, −1 ≤ sin1/x ≤ 1, x ≠ 0

⇒ −x^{2} ≤ x^{2 }sin1/x ≤ x^{2}

⇒ lim_{x→0} (−x^{2}) ≤ lim_{x→0} (x^{2 }sin1/x) ≤ limx_{2x→0}

⇒ 0 ≤ lim_{x→0} (x^{2 }sin1/x) ≤ 0

⇒ lim_{x→0} (x^{2 }sin1/x) = 0

⇒ lim_{x→0−} f(x) = 0

Similarly,

lim_{x→0+} f(x) = lim_{x→0+} (x^{2 }sin1/x)

= lim_{x→0} (x^{2 }sin1/x) = 0

⇒ lim_{x→0−} f(x) = f(0) = lim_{x→0+} f(x)

Therefore, f is continuous at x = 0.

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 24

## Determine if f defined by f(x) = {(x^{2 }sin1/x, if x≠0) (0, if x=0) is a continuous function?

**Summary:**

Function defined by f(x) = {(x^{2 }sin1/x, if x ≠ 0) (0, if x = 0) is a continuous function. From the above observations, it can be concluded that f is continuous at every point of the real line

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