Is the density a continuous function?

Question

Let $X$ be a continuous random variable with density $$f_X(x)=x^2 I_{(0,1]}(x)+\left(\frac{7}{4}-\frac{3}{4}x\right)I_{(1,7/3)}(x)$$

The density function is

$(A) \text{ right constant, but not continuous. }\\ (B) \text{ piecewise constant. }\\ (C) \text{ continuous. }\\ (D) \text{ left continuous, but not continuous.}$

Would it be correct to say that it is continuous?

I know that the density is equal to $x^2$ when $x\in (0,1]$ and it's equal to $\left(\frac{7}{4}-\frac{3}{4}x\right)I_{(1,7/3)}(x)$ when $x\in (1,7/3)$

So, the function would look like this on the interval $(0,7/3)$:

Answer 1

Yes, it is continuous.

Polynomials are continuous.

It remains to check that

$$\lim_{x \to 1^-}f(x) = \lim_{x \to 1^+}f(x)=f(1)$$ of which it is clear that it holds with value $1$.