# NCERT Class 12-Mathematics: Chapter –5 Continuity and Differentiability Part 3

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**Question 4:**

Show that the function defined by

is continuous at .

**Answer:**

Left hand limit at is given by

Similarly

Thus, Hence is continuous at

**Question 5:**

Given .Find the points of discontinuity of the composite function .

**Answer:**

We know that is discontinuous at

Now, for ,

which is discontinuous at .

Hence, the points of discontinuity are and .

**Question 6:**

Let , for all . Discuss the derivability of

**Answer:**

We may rewrite

Now,

Now

Since the left hand derivative and right hand derivative both are equal, hence is differentiable at .

**Question 7:**

Differentiate

**Answer:**

Let . Using chain rule, we have

**Question 8:**

**Answer:**

Given . differentiating both sides w.r.t. , we have

Or

Therefore,

**Question 9:**

If , prove that

**Answer:**

Given that . Differentiating both sides , we have

Or

Which implies that