Question

If f(x) is continuous and g(x) is discontinuous function then f(x) + g(x) will be?
(a) Continuous function
(b) Discontinuous function
(c) Constant function
(d) Identity function

Answer 1

Hint: First we need to learn what are continuous and discontinuous functions. In the layman terms: The graph of a continuous function can be drawn without lifting the pencil from the paper, and for discontinuous functions, we cannot do so. To solve this problem we can take an example of two functions one of which is continuous and one of which is discontinuous then we will add both of them and analyze the resultant function.

Complete step-by-step solution:
We know that for a function to be continuous at a point, the function must exist at the point and any small change in x produces only a small change in f(x) and Many functions have discontinuities (i.e. places where they cannot be evaluated) and hence they are called discontinuous functions.
In the layman terms: The graph of a continuous function can be drawn without lifting the pencil from the paper, and for discontinuous functions, we cannot do so.
For example,
f(x) = x ….(1)
is a continuous function as the above function can be drawn on the paper without lifting the pencil, and any small change in the value of x is producing only a small change in the value of f(x).
And the function,
g(x) = $\left\{ \begin{align}
  & 5\,\,\,x=0 \\
 & 0\,\,\,x\ne 0 \\
\end{align} \right.\,\,\,\,.....(2)$
is a discontinuous function as when we will try to draw this we will need to lift our pencil at x = 0 as there is sudden change in the value of the function g(x).
Now, if we add both the functions f(x) + g(x), we get
f(x) + g(x) = $\left\{ \begin{align}
  & 5\,\,\,x=0 \\
 & x\,\,x\ne 0 \\
\end{align} \right.$
Now, if we will try to draw this function on the graph we will see that there will be a sudden change in the value of x, it will suddenly increase from 0 to 5 hence it is a discontinuous function.
From the above example we have observed that if f(x) is a continuous function and g(x) is a discontinuous function then, their sum f(x) + g(x) will be a discontinuous function.

Note: You can also solve this question by taking any other example of continuous and discontinuous function you like and will get the same result.
You should remember this for the future problems that the addition or subtraction of two continuous functions will always be continuous whereas the addition and subtraction of continuous and a discontinuous function will be a discontinuous function.